WARPED CONES, (NON-)RIGIDITY,
AND PIECEWISE PROPERTIES
DAMIAN SAWICKI

arXiv:1707.02960v1 [math.MG] 10 Jul 2017

With an appendix by Dawid Kielak and Damian Sawicki
Abstract. We prove that if a quasi-isometry of warped cones is induced by
a map between the base spaces of the cones, the actions must be conjugate by
this map. The converse is false in general, conjugacy of actions is not suﬃcient
for quasi-isometry of the respective warped cones. For a general quasi-isometry
of warped cones, using the asymptotically faithful covering constructed in a
previous work with Jianchao Wu, we deduce that the two groups are quasiisometric after taking Cartesian products with suitable powers of the integers.
Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties
of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an
application, we calculate the asymptotic dimension of a warped cone and improve bounds by Szabo, Wu, and Zacharias and by Bartels on the amenability
dimension of actions of virtually nilpotent groups.
In the appendix, we justify optimality of our result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasiisometric groups, contrary to the case of box spaces.

1. Introduction
Finitely generated groups provide interesting examples of metric spaces to study
by geometers. In particular, the first explicit construction of expanders, due to
Margulis, is done by taking a family of finite quotients of a group with relative
property (T). This approach was formalised by J. Roe [23], who called such a
family of finite quotients a box space. One of the main ideas of geometric group
theory is to study groups via their actions. For a group action on a compact space,
Roe introduced another construction called a warped cone [24]. In fact, the latter
construction generalises the former [27].
For box spaces, one can use the ambient group to prove some properties of the
box space and vice versa. In particular, amenability of the group is equivalent
to property A of its box space [23] and the Haagerup property of the group is
equivalent to admitting a fibred coarse embedding into a Hilbert space by the box
space [7, 8, 23]. Furthermore, every quasi-isometry of box spaces lifts to a quasiisometry of groups [14].
Since a warped cone OΓ Y comes from an action ΓyY of a group, the group itself
is not the appropriate (sufficient) object to be compared with a warped cone (as
opposed to the above case of box spaces). An appropriate space was constructed
recently by Jianchao Wu and the author [30]. Its usefulness comes from the fact
that there is an asymptotically faithful (see Definition 2.3) quotient map from it to
the warped cone, just like the sequence of quotient maps from the group onto the
box space is asymptotically faithful.
Key words and phrases. Warped cone; asymptotically faithful covering; quasi-isometry; orbit
equivalence; asymptotic dimension; continued fractions.
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DAMIAN SAWICKI

Results. Motivated by asymptotically faithful maps we introduce piecewise versions of known properties of metric spaces (like hyperbolicity, asymptotic dimension, property A, or coarse embeddability into Banach spaces). On the theoretical
side, it enables us to characterise certain properties of groups by piecewise versions
of these properties for box spaces (Corollary 6.9) or warped cones (Proposition
7.4). The above-mentioned characterisations from [7, 8, 23] concerned “equivariant”
properties of groups, that is, taking into account their group structure, while our
characterisations concern metric properties. For brevity, we omit some assumptions
in formulations of theorems below. They are all valid for warped cones over isometric free actions on nice spaces, including manifolds and ultrametric spaces (e.g.
profinite completions metrised as in [27]).
Theorem A. Let P be one of the following properties: “property A”, “asymptotic
dimension at most k” or “admitting a coarse embedding into E”, where k ∈ N and
E is a Banach space. The following are equivalent:
• a group Γ has property P ,
• a box space (Γ/Γi )i satisfies property P piecewise,
• a warped cone (tY, dΓ )t satisfies property P piecewise.
On the quantitative side, we calculate the asymptotic dimension of a warped
cone over the action ΓyY , using the asymptotic dimension of Γ and the topological
dimension of Y (Theorem 7.1).
Theorem B. If asdim OΓ Y < ∞, then asdim OΓ Y = asdim Γ × Zm , where m is
the topological dimension of Y .
Consequently, we have the following dynamical result (Corollary 8.5) about equivariant asymptotic dimension (also known as amenability dimension), which builds
on and strengthens the estimates by Szabo, Wu, and Zacharias [32] and by Bartels
[3]. This was a problem raised by Willett.
Theorem C. If Γ is a virtually nilpotent group and Y is a compact metrisable
space, then eq-asdim(ΓyY ) ≤ asdim Γ + dim Y .
The result that quasi-isometry of box spaces implies quasi-isometry of groups [14]
was recently strengthened to the fact that it implies commensurability of groups
[10]. It leads to the following question.
Question D. Does a quasi-isometry OΓ Y ≃ O∆ X imply isomorphism / commensurability / quasi-isometry of Γ and ∆?
In Appendix A, we give a negative answer to this question. For instance, there
exist free actions Z3 yT1 and Z2 yT2 yielding quasi-isometric warped cones (after
passing to a subsequence). In fact, there exist (continuum many) free actions of Zd
for any d ∈ N yielding warped cones quasi-isometric to ones over trivial actions. We
also construct actions of the infinite dihedral group yielding warped cones quasiisometric to the collection of spheres over all positive radii.
Nonetheless, quasi-isometry of warped cones implies the following “stable” quasiisometry (Theorem 3.1), which was very recently obtained independently by de Laat
and Vigolo [16]:
Theorem E. OΓ M ≃ O∆ N =⇒ Γ × Zm ≃ ∆ × Zn for manifolds M, N of
dimension m and n respectively.
Since the study of warped cones is the study of group actions, one can ask about
the case when a quasi isometry OΓ Y ≃ O∆ X is induced by a map f : Y → X. We
have the following result (Theorem 4.1), which yields a very strong positive answer
to Question D.

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3

Theorem F. If a homeomorphism of connected spaces f : Y → X induces a quasiisometry OΓ Y ≃ O∆ X, then in fact Γ and ∆ are isomorphic and the actions are
conjugate by f .
In Section 4 we also study more general cases, when f is not required to be a
homeomorphism and spaces Y and X are not connected.
However, we offer an example showing that a topological conjugacy of actions is
not sufficient for quasi-isometry of warped cones (Theorem 5.3).
Theorem G. There exist a Γ-space Y with two invariant metrics d and d′ , such
that OΓ (Y, d) does not admit a coarse embedding into a Hilbert space and OΓ (Y, d′ )
does.
2. Definitions
Throughout this note, Γ will be a discrete finitely generated group and S will be
a finite symmetric generating set for Γ.
Definition 2.1 (J. Roe [24]). Let (Y, d) be a compact metric space with an action
of Γ = hSi. The warped cone is the collection of metric spaces OΓ Y = ({t} × Y )t>0
(denoted (tY )t>0 for brevity), where each set tY is equipped with the largest metric
dΓ such that:
dΓ (tx, ty) ≤ d(tx, ty)

and

dΓ (ty, s(ty)) ≤ 1 for any s ∈ S,

where d(tx, ty) = td(x, y).
The family (tY, d)t>0 will also sometimes be used. It is called the open cone (or
the infinite cone) and denoted by OY .
Note that the original
S definition [24] gives a metric on the unified warped cone
OΓu Y = (0, ∞) × Y = t>0 tY . However, some properties make sense only for a
family of levels tY , rather than the metric space OΓu Y build from them [29, 34].
For some other properties, a property holds uniformly across the levels if and only
if it holds for OΓu Y [27, Lemma 4.1 and Proposition 5.2], but working with levels
simplifies notation a bit.
Some results in this paper can be formulated and proved for the unified warped
cone as well. However, for example in Theorem E, the result would be slightly
weaker: instead of the quasi-isometry Γ × Zm ≃ ∆ × Zn one would get Γ × Zm+1 ≃
∆ × Zn+1 .
Definition 2.2. Let X and Y be two families of metric spaces.
• They are quasi-isometric if there exists an index set I with surjective maps
I ∋ i → Xi ∈ X and I ∋ i → Yi ∈ Y and a family of functions fi : Xi → Yi
which are quasi-isometries in a uniform way. That is, there exists constants
C ≥ 1 and A ≥ 0 not depending on i such that for every i ∈ I and every
x, x′ ∈ Xi we have
C −1 d(x, x′ ) − A ≤ d(fi (x), fi (x′ )) ≤ Cd(x, x′ ) + A

and A-neighbourhood of im(fi ) equals Yi .
• Similarly, X embeds coarsely into Y if there exists an index set I with a
surjective map I ∋ i → Xi ∈ X and any map I ∋ i → Yi ∈ Y and a family
of functions fi : Xi → Yi that are coarse embeddings in a uniform way.
That is, there exist two non-decreasing and converging to infinity functions
ρ− , ρ+ : [0, ∞) → [0, ∞) such that for every i ∈ I and x, x′ ∈ Xi we have
ρ− ◦ d(x, x′ ) ≤ d(fi (x), fi (x′ )) ≤ ρ+ ◦ d(x, x′ ).

• For brevity, we will call such families of maps f = (fi ) respectively a quasiisometry or a coarse embedding and write f : X → Y.

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DAMIAN SAWICKI

We will use the wording “(C, A)-quasi-isometry” to specify constants.
As we emphasised in Section 1, in addition to coarse embeddings and quasiisometries, we will make heavy use of maps that are asymptotically faithful.
Definition 2.3 (Willett–Yu [35]). Let (I, ≤) be a linearly ordered set and let
(fi : Xi → Yi )i∈I be a family of surjective maps. It is called asymptotically faithful
if for every R ∈ N there exists j ∈ I such that for every i ≥ j and every x ∈ Xi the
restriction fi : B(x, R) → B(fi (x), R) is an isometry. Then, we call fi an R-local
isometry.
Practically, we will only use I = N or I = (0, ∞) with their standard orders. The
respective definition for a single surjective map f : X → Y would say that for every
R there is a bounded set B ⊆ Y such that we have the above isometric condition
for x ∈
/ f −1 (B).
Examples. One family of examples of spaces with interesting asymptotically faithful coverings appearing in the literature are box spaces (Γ/Γi )i , where fi is the
quotient map Γ → Γ/Γi . Here, Γ is a finitely generated residually finite
T group and
Γi are its finite index normal subgroups such that Γi ⊇ Γi+1Fand i Γi = {e}.
Typically, one defines a box space as a coarse disjoint sum ( Γ/Γi , d), where d
restricted
F to every “component” Γ/Γi is equal to the quotient metric on Γ/Γi and
d(Γ/Γj , i6=j Γ/Γi ) → ∞ as j → ∞ (all such metrics are coarsely equivalent). With
F
this definition, we can say that the quotient map f : {2i | i ∈ N} × Γ → i Γ/Γi is
asymptotically faithful.
However, there is no canonical way to define distances between different components of a box space, so it makes no sense to talk canonically about quasi-isometries
of box spaces. This is one of the reasons why the author F
prefers to define a box
space as
a
sequence.
Another
reason
is
that
for
a
box
space
i Γ/Γi the double box
F
space i Γ/Γ⌈i/2⌉ is not coarsely equivalent to it (at least when the original box
space does not have repetitions itself). In fact, one can obtain continuum many
non-equivalent box spaces in this rather trivial way. This is no longer true for our
definitions, the double box space {Γ/Γi }i ⊔{Γ/Γi }i is quasi-isometric to the original
box space {Γ/Γi }i .
Another family of examples is given by sequences of graphs (Gi ) with increasing
e i → Gi , where G
ei
girth (that is, length of the shortest simple loop) and maps fi : G
e
is the universal cover of Gi and fi is the covering map. Note that Gi is a tree, and
e i are isometric if vertices of graphs Gi have a fixed degree (independent from
all G
i and a particular vertex).
Any box space of a free group (with the graph structure coming from its standard
generators) fits in both categories.
A new class of examples was found in [30, Lemmas 3.6 and 3.8], where it was
proved that a suitably metrised product Γ × tY provides an asymptotically faithful
covering for a warped cone via the map (γ, ty) 7→ γ(ty) if and only if the continuous
action ΓyY is free.
Theorem 2.4 (Sawicki–Wu [30]). Let Γy(Y, d) be a continuous action. Consider
the space Γ × tY with the largest metric d1 such that
d1 ((γ, ty), (η, ty)) ≤ |γη −1 | and d1 ((γ, ty), (γ, ty ′ )) ≤ td(γy, γy ′ ).

Then, (tY, dΓ ) is the quotient space of Γ × tY under the quotient map π(γ, ty) =
γ(ty).
Moreover, the family of quotient maps (π : (Γ × tY, d1 ) → (tY, dΓ ))t>0 is asymptotically faithful if and only if the action ΓyY if free.
We will also often need the following results.

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5

Lemma 2.5 ([27, Proposition 2.1]). Suppose Γy(Y, d) is Lipschitz and L ≥ 1 is
the Lipschitz constant for the action of the generators s ∈ S of Γ. Let
DΓ (ty, ty ′ ) = inf (|γ| + td(γy, y ′ )).
γ∈Γ

Then the following inequality holds for all y, y ′ ∈ Y :

′

dΓ (ty, ty ′ ) ≤ DΓ (ty, ty ′ ) ≤ LdΓ (ty,ty ) · dΓ (ty, ty ′ ).
In particular, when the action is isometric, then D = d.
Lemma 2.6 ([27, Remark 3.1]). Let Γy(Y, d) be a continuous action. Then
(
∞,
if Γy 6= Γy ′ ,
lim d(ty, ty ′ ) =
t→∞
min{|η| | y ′ = ηy}, otherwise.
In the latter case, d(ty, ty ′ ) equals the limit for sufficiently large t.
3. Quasi-isometry of cones implies stable quasi-isometry of groups
Recall that by an observation of Khukhro and Vallette [14], quasi-isometric box
spaces must come from quasi-isometric groups. For finitely presented groups, this
was recently strengthened by Delabie and Khukhro [10] – if two box spaces are
quasi-isometric, the ambient groups must share a finite index
F normal subgroup. In
fact, roughly speaking, from the geometry of the box space Γ/Γi one can recover
the normal subgroups Γi , which shows that box spaces are very rigid.
In Appendix A, we show that this is no longer true for warped cones – there exist
quasi-isometric warped cones over actions of non-quasi-isometric groups (moreover,
on non-homeomorphic spaces). Nonetheless, we have the following result yielding
“stable” quasi-isometries.
Theorem 3.1. Let Γ, ∆ be two finitely generated groups and M, N be two compact
Riemannian manifolds of dimension m and n respectively with free isometric actions
ΓyM and ∆yN . Then, quasi isometry of warped cones OΓ M ≃ O∆ N implies
that there is a quasi-isometry of groups Γ × Zm ≃ ∆ × Zn .
Proof. Take a sequence (ti ) ⊆ (0, ∞) going to infinity and let (τi ) be another
sequence such that (fi : (ti M, dΓ ) → (τi N, d∆ ))i∈N is a (C, A)-quasi-isometry for
some C ≥ 1 and A ≥ 0. Since the diameter of (ti M, dΓ ) goes to infinity, the
diameter of (τi N, d∆ ) also goes to infinity and hence the sequence (τi ) goes to
infinity. Let R ∈ N and let i = i(R) be sufficiently large (to be specified later).
Let Γ × ti M be equipped with the following product metric:
d1 ((γ, ti x), (γ ′ , ti x′ )) = |γ ′ γ −1 | + ti d(x, x′ )

and similarly for ∆ × τi N . Note that the ball of radius R in d1 about (γ, ti x) (and
similarly for (δ, τi y)) is contained in the product
B(γ, R) × B(ti x, R) = B(γ, R) × B(x, R/ti ),
where we identified ti M and M . For ti sufficiently large the following map πΓ :
(Γ × ti M, d1 ) ∋ (γ, ti x) 7→ γ(ti x) ∈ (ti M, dΓ ),
when restricted to the above product, is an isometry by Theorem 2.4.
Now, since map fi is a (C, A)-quasi-isometry, the image of any ball B(ti x, ρ)
in (ti M, dΓ ) under fi is contained in the ball B(f (ti x), Cρ + A) and moreover
A-neighbourhood of this image contains B(f (ti x), ρ/C − 2A). By increasing i if
necessary, we can assume that the map π∆ : (δ, τi y) 7→ δ(τi y) is an isometry on

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DAMIAN SAWICKI

B((e, f (ti x)), CR + A). Hence, we can consider a local inverse σ∆ of π∆ and obtain
a map ji = σ∆ ◦ fi ◦ πΓ :
ji : B((e, ti x), R) → B((e, f (ti x)), CR + A),
which is a (C, A)-quasi-isometric embedding such that for any ρ ≤ R the Aneighbourhood of the image of the ball B((e, ti x), ρ) contains B((e, f (ti x)), ρ/C −
2A).
Let r > 0 be a positive number such that the exponential maps expx : Tx M ⊇
B(0, r) → B(x, r) and expy : Ty N ⊇ B(0, r) → B(y, r) are homeomorphisms and
that all expx , expy and their inverses are L-Lipschitz maps with L not depending
on x ∈ M and y ∈ N . We can also assume that these maps preserve spheres, that
is, d(0, z) = d(exp∗ (0), exp∗ (z)) for any z ∈ B(0, r) and ∗ ∈ M ⊔ N .
For i sufficiently large, we can assume that r > R/ti and r > (CR + A)/τi .
Consider the composition:
−1
ki = (id∆ ×(τi ◦ exp−1
f (ti x) )) ◦ j ◦ (idΓ ×(expx ◦ti )),

where t−1
denotes the multiplication operator in Tx M and τi denotes the multiplii
cation operator in Tf (ti x) N (we keep identifying points in ti M and M , and points
in τi N and N ). Map ki is well defined on the ball B((e, 0), R) inside Γ × Rm
(with the ℓ1 -metric) and is a (CL2 , AL)-quasi-isometric embedding into the ball
B((e, 0), CR + A) inside ∆ × Rn . Moreover, for any ρ ≤ R the AL-neighbourhood
of the image of the ball B((e, 0), ρ) contains the ball B((e, 0), ρ/C − 2A).
Now, we can pre-compose map ki with a quasi-isometry between B((e, 0), R)
and B((e, 0), R) ∩ (Γ × Zm ) and post-compose with a quasi-isometry between balls
B((e, 0), CR + A) and B((e, 0), CR + A) ∩ (∆ × Zn ). Let us keep the old notation
′
′
for these new map and new
for
√ balls.√Note that now it is a (C , A )-quasi-isometry
′
2
′
2
C = CL and A = 2CL m+2 n+AL and for any ρ ≤ R the A′ -neighbourhood
of the image of B((e, 0), ρ) ⊆ Γ × Zm contains B((e, 0), ρ/C − 2A) ⊆ ∆ × Zn .
Now, let U be any non-principal ultrafilter on N. We define
k : Γ × Zm → ∆ × Zn by k(γ, z) = lim ki(R) (γ, z).
R→U

First, observe that for a given (γ, z) the value ki (γ, z) is defined for sufficiently large
i, so the limit makes sense. Furthermore, since ki (e, 0) = (e, 0) for all i and all ki are
(C ′ , A′ )-quasi-isometric embeddings, we know that ki (γ, z) always belongs to the
finite set B((e, 0), C ′ d((γ, z), (e, 0)) + A′ ), and hence the limit exists. Obviously, it
is still a (C ′ , A′ )-quasi-isometric embedding. Furthermore, since A′ -neighbourhood
of ki (B((e, 0), ρ)) contains B((e, 0), ρ/C − 2A), the same holds for k. Since ρ is
arbitrary, map k is a quasi-isometry.

Remark 3.2. Note that we do not use in the proof that the whole warped cones
are quasi-isometric. It suffices that there is an unbounded sequence (ti )i ⊆ R such
that (ti M, dΓ )i is quasi-isometric to (τi N, d∆ )i for some (necessarily unbounded)
sequence (τi )i .
Remark 3.3. Similarly, one can prove that if a box space (Γ/Γi )i is quasi-isometric
to (a subsequence of ) a warped cone (τi N, d∆ )i for an action ∆yN as in Theorem 3.1, then Γ ≃ ∆ × Zn . In this case, a box space of a group Γ cannot be
quasi-isometric to a warped cone over its action, unless Γ ≃ Γ × Zn .
However, this is restricted to actions on manifolds, because the author showed in
[27] that any box space (Γ/Γi )i may be realised as a sequence of levels (τi Y, dΓ )i for
Y = lim Γ/Γi .
←−
By similar arguments, one obtains the following.

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7

Corollary 3.4. Let Γ, ∆ be two finitely generated groups and M, N be two compact
Riemannian manifolds of dimension m and n respectively with free isometric actions
ΓyM and ∆yN . Then, if the warped cone OΓ M embeds coarsely into O∆ N , then
there is a coarse embedding of groups Γ × Zm → ∆ × Zn .
Remark 3.5. Note that in the proof of Theorem 3.1 we used only neighbourhoods
of one orbit of the action ΓyM . Hence, with a more careful treatment, one can
relax the assumption of the freeness of ΓyM to the existence of one free orbit. The
same applies to Corollary 3.4.
Tim de Laat and Federico Vigolo [16] very recently obtained independently a
different straightening of Theorem 3.1, in which they assume only that the actions
ΓyM and ΛyN are essentially free.
4. Quasi-isometry induced by one map
The lack of rigidity exhibited in the Appendix A and in the formulation of
Theorem 3.1 and Corollary 3.4 comes from the fact that a quasi-isometry of warped
cones need not respect the equivalence relation induced by the action (it may ignore
the division into the Γ- and ∆-coordinate and the M - and N -coordinate (or Zm and Zn -coordinate), that one has in Γ × ti M and ∆ × τi N (or Γ × Zm and ∆ × Zn ).
However, if it is induced by a single continuous map M → N , we have the following
rigidity behaviour.
Theorem 4.1. Let f : Y → X be a continuous map from a connected space, and
let ΓyY contain a free orbit and ∆yX be free. If f induces a quasi-isometry of
warped cones OΓ Y and O∆ X, then
• there is a short exact sequence 1 → F → Γ → ∆′ → 1, where ∆′ , the
stabiliser of im(f ) inside ∆, is of finite index in ∆, and F is finite;
• there is a homeomorphism h : im f → Y /F such that the quotient map
q : Y → Y /F equals h ◦ f and h conjugates the actions ∆′ y im f and
Γ/F yY /F ;
• im f is one of [∆ : ∆′ ] many connected components of X and ∆ acts on
them transitively.
In particular, if f is a homeomorphism, then Γ and ∆ are isomorphic and the
actions are conjugate by f .
Proof. Recall from Lemma 2.6 that limt→∞ dΓ (ty, ty ′ ) equals ∞ if y and y ′ lie in
different orbits and equals
min{|η| | η ∈ Γ, ηy = y ′ }

(1)

if Γy ′ = Γy. By the assumption, for every t > 0 there is τ (t) such that
(2)

C −1 dΓ (ty, ty ′ ) − A ≤ d∆ (τ (t)f (y), τ (t)f (y ′ )) ≤ CdΓ (ty, ty ′ ) + A.

We conclude that

lim d∆ (τ f (y), τ f (y ′ )) = lim d∆ (τ (t)f (y), τ (t)f (y ′ )) < ∞

τ →∞

t→∞

if and only if limt dΓ (ty, ty ′ ) < ∞ and hence f preserves orbits, that is,

(3)

and also

f (Γy) = ∆f (y) ∩ im(f ),

Γy 6= Γy ′ =⇒ ∆f (y) 6= ∆f (y ′ ),
meaning that f is injective as a map Y /Γ → X/∆.
Fix y ∈ Y . We will prove that δ(Γ, y) is coarsely dense in ∆. Take x = f (y), any
δ ∈ ∆, and put x′ = δx. By the assumption, for every n ∈ N there is xn ∈ im(f )
such that d∆ (nx′ , nxn ) ≤ A. In particular, for any m ≤ n we have d∆ (mx′ , mxn ) ≤

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DAMIAN SAWICKI

A. By compactness, a subsequence of xn has a limit x0 ∈ im(f ) which satisfies
d∆ (mx′ , mx0 ) ≤ A for all m ∈ N. In particular x′ = δ0 x0 for some |δ0 | ≤ A. This
means that x and x0 lie in the same ∆-orbit, and hence, by (3), there is some
γ ∈ Γ such that δ(γ, y)x = x0 and consequently δx = x′ = δ0 x0 = δ0 δ(γ, x)x. In
particular
(4)

δ = δ0 δ(γ, y).

Assume now that ΓyY has a free orbit and let y0 ∈ Y be an element of such an
orbit. Then, formula (1) simplifies to limt dΓ (ty0 , γty0 ) = |γ|, which, together with
(2) yields
(5)

|δ(γ, y0 )| ≥ C −1 |γ| − A.

Now, since the ∆-action is free, for every γ ∈ Γ and y ∈ Y there is a unique
δ(γ, y) such that f (γy) = δf (y). We will show that this function is continuous.
Due to (1) and (2), we know that
(6)

|δ(γ, y)| ≤ C|γ| + A,

in particular δ(γ, ·) is a function Y → B(e, C|γ|+ A) into a finite set. To confirm its
continuity, it suffices to observe that the inverse image of a point is always closed.
Now recall that we assumed that Y is connected. A continuous function δ(γ, ·)
from a connected space to a discrete space must be constant. Note that by definition
δ is a cocycle, that is, it satisfies:
δ(γ2 , γ1 y)δ(γ1 , y) = δ(γ2 γ1 , y),
but as there is no dependence on y, it simply means that δ(γ) ..= δ(γ, y) is a
homomorphism. In particular (5) means that the kernel F of δ is finite, as contained
in the ball B(e, CA). Similarly, (4) (where |δ0 | ≤ A) is equivalent to saying that
the subgroup δ(Γ) is of finite index in ∆.
Note that δ(γ, y) = e if and only if f (γy) = f (y), but, since δ does not depend
on the second coordinate, we have γ ∈ F = ker δ if and only if f (γy) = f (y)
for some (equivalently: all) y. Hence, f is well-defined as a map Y /F → im f .
For the same reason, f restricted to any Γ-orbit in Y /F is injective. In fact, it is
injective on Y /F . Indeed, assume that f (y) = f (y ′ ) for some y, y ′ with Γy 6= Γy ′ .
Then limt d(ty, ty ′ ) = ∞ and, by (2), limt d(τ (t)f (y), τ (t)f (y ′ )) = ∞, which by (1)
implies that ∆f (y) 6= ∆f (y ′ ).
Since Y is compact, f : Y /F → im f is a homeomorphism. By (3) for y ∈ Y
we now that for any δ ′ ∈ ∆ such that δ ′ f (x) ∈ im f , there is γ ∈ Γ such that
f (γy) = δ ′ f (x). But then δ(γ) = δ ′ , which shows that all δ ′ stabilising im(f ) are in
the image of δ. The converse inclusion is obvious. The above reasoning also shows
that if δ im(f ) 6= im(f ), then δ im(f )∩im(f ) = ∅. Since ∆′ has finite index, say k in
∆, there exist δ1 , . . . δk ∈ ∆ \ ∆′ such that {e, δ1 , . . . , δk } im(f ) = ∆ im(f ). Hence
Sk
∆ im(f ) \ im(f ) = i=1 δi im(f ) a closed set, so im(f ) is a clopen in ∆ im(f ), hence
a connected component.
By compactness of ∆ im(f ), if there exists x ∈ X \∆ im(f ), then a similar reasoning to the one in the proof of Lemma 2.6 shows that limτ →∞ dΓ (τ x, τ ∆ im(f )) = ∞,
which would contradict the fact that A-neighbourhood of im(f ) is the whole τ (t)Y .
Hence X = ∆ im(f ).
The proof is finished in the case when f is continuous. If it is additionally a
homeomorphism, then F must be trivial, as f factors through Y /F . By surjectivity
of f , (3) and the freeness of the action of ∆, δ(·, y) must be surjective. Hence δ is
an isomorphism.

Remark 4.2. Recall that two actions ΓyY and ∆yX are orbit equivalent if there
is an “isomorphism” Y → X satisfying (3). Typically, one considers probability

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

9

measure preserving actions, Borel isomorphisms and (3) is required up to measure 0
[33]. See also Remark 4.6.
Remark 4.3. In Theorem 3.1, if f is assumed to be injective, then, instead of a
free Γ-orbit, it suffices to assume faithfulness of ΓyY .
Proof. Note that the only place in the proof of Theorem 3.1 where we used a free
orbit of Γ is (5), which was later used to conclude that δ has a finite kernel. However,
if f is injective, we can conclude that already from the faithfulness of ΓyY . Indeed,
for every γ 6= e, there is some yγ ∈ Y with γyγ 6= yγ and, consequently, f (γyγ ) 6=
f (yγ ). In particular δ(γ) = δ(γ, yγ ) 6= e, so δ is a monomorphism.

By a similar argument, one can obtain:
Corollary 4.4. Let f : Y → X be a continuous map from a connected space, and
let ΓyY contain a free orbit and ∆yX be free. If f induces a coarse embedding of
warped cones OΓ Y → O∆ X then ∆ contains a subgroup isomorphic to a quotient
of Γ by a finite index subgroup.
Without the connectedness assumption, one obtains a cocycle instead of a homomorphism.
Corollary 4.5. If both actions are free and a homeomorphism f : Y → X induces a
quasi-isometry of the respective warped cones, one gets a continuous cocycle δ : Γ ×
Y → ∆ (which is bijective as a function Γ → ∆ for a fixed y ∈ Y ) satisfying
(7)

C −1 |γ| − A ≤ |δ(γ, y)| ≤ C|γ| + A

for some constants C ≥ 1 and A ≥ 0. In particular Γ and ∆ are bijectively quasiisometric.
If f is only injective or surjective, we get the appropriate properties of the cocycle
and the quasi-isometry.
Proof. The inequality (7) is just a combination of (6) and (5) (now (5) holds for
any y as all orbits are free).
Note that the map δy = δ(·, y) : Γ → ∆ is injective by freeness of the Γ-action
and injectivity of f and it is surjective by (3), surjectivity of f and freeness of the
∆-action. If we do not assume surjectivity of f , then still some A-neighbourhood
of δy (Γ) is the whole ∆ by the same argument as in the proof of Theorem 3.1.
Moreover, δy is a quasi-isometric embedding. Indeed,
d(δy (γ), δy (η)) = |δy (γ)δy−1 (η)|

= |δ(γ, y)δ −1 (η, y)|

= |δ(γ, y)δ(η −1 , ηy)|

= |δ(γη −1 , ηy)|

∈ [C −1 |γη −1 | − A, C|γη −1 | + A]

= [C −1 d(γ, η) − A, Cd(γ, η) + A].



Remark 4.6. In some cases one can deduce conjugacy of actions from the continuous orbit equivalence (established by Corollary 4.5), that is, from the existence of
an orbit preserving homeomorphism f as above and continuity of the cocycle δ and
the “opposite” cocycle γ : ∆ × X → Γ, but in general this is a weaker condition than
conjugacy [17].
The continuity of f leads to continuity of δ and, if Y is connected, to a homomorphism. However, if f is not continuous, we still get the following.

10

DAMIAN SAWICKI

Corollary 4.7. If both actions are free and a bijection f : Y → X induces a quasiisometry of the respective warped cones, one gets a cocycle δ : Γ × Y → ∆ (bijective
for a fixed y ∈ Y ) satisfying C −1 |γ| − A ≤ |δ(γ, y)| ≤ C|γ| + A for C ≥ 1 and
A ≥ 0. In particular Γ and ∆ are bijectively quasi-isometric.
Remark 4.8. Assume that Y, X admit invariant ergodic measures, the actions are
free up to measure 0, and f , inducing a quasi-isometry OΓ Y ≃ O∆ X, is a measure
space isomorphism (sometimes it suffices to assume that on subsets of Y and X of
positive measure). In this case δ is a measurable map (and the actions are orbit
equivalent by Remark 4.2) and we can use various “orbit equivalence (super)rigidity
theorems” to conclude that the respective actions are conjugate by f (in particular,
that Γ and ∆ are isomorphic).
For example, the following was obtained in [22].
Theorem 4.9 (Popa). Let Γy(Y, µ) be a Bernoulli action of a w-rigid group with
no finite normal subgroups. Assume that there is a free ergodic action Λy(X, ν) and
an orbit-preserving measure space isomorphism f : Y → X. Then Λ is isomorphic
to Γ and f establishes a conjugacy of actions.
Above, Bernoulli action is an action Γy(X0 , µ0 )Γ on X0 -valued functions on Γ
by shifts, i.e. (γf )(η) = f (γ −1 η), where (X0 , µ0 ) is some non-atomic probability
space and (X0 , µ0 )Γ is equipped with the product measure; w-rigidity means that
Γ admits an infinite normal subgroup Γ′ such that (Γ, Γ′ ) has relative property
(T). Freeness, orbit-preserving, as well as conjugacy should be understood up to
measure zero, and in fact f may be assumed to be a map from a subset of X of
positive measure. See the original paper [22] or a survey [33] for details and more
general statements, as well as references to other results of this kind.
The following is a version of Corollaries 4.5 and 4.7 for coarse embeddings.
Corollary 4.10. If both actions are free and a (continuous) map f : Y → X induces
a coarse embedding OΓ Y → O∆ X, one gets a (continuous) cocycle δ : Γ × Y → ∆
satisfying
(8)

ρ− (|γ|) ≤ |δ(γ, y)| ≤ ρ+ |(γ|)

for some non-decreasing functions ρ± : [0, ∞) → [0, ∞) going to infinity. In particular, Γ embeds coarsely into ∆.
Example 4.11. Below, we present some examples of quasi-isometric warped cones
over non-conjugate actions.
(1) It is not difficult to observe that for an isometric action of a finite group
F yY , the quotient map f : Y → Y /F induces a quasi-isometry between
OF Y and O{1} (Y /F ) (in fact an almost isometry with the additive constant
diam F ).
This shows that it is reasonable to consider continuous maps and not
only homeomorphisms in the results in this section and that group F is
necessary in the statement of Theorem 4.1.
(2) The above example can be easily generalised. Assume that there is a finite
subgroup F ⊳ Γ whose action F yY is isometric. Then the quotient map
f : Y → Y /F induces a quasi-isometry between OΓ Y and OΓ/F (Y /F ).
(3) In the above examples ∆ is a quotient of Γ (that is, ∆ = ∆′ ) and f is
surjective. The simplest example showing that that does not need be the case
is when Γ and ∆ are finite groups acting on themselves by left multiplication.
All warped cones over such actions are quasi-isometric to the one for Γ =
{1} and the quasi-isometry is induced by the inclusion Γ < ∆.

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

11

Note that by the last bullet point in Theorem 4.1, if we want ∆′ to be a
proper subgroup of ∆, Y is connected, and ∆yX is free, then the space X
must be disconnected.
(4) Another possibility is to drop the assumption of freeness of ∆yX. Consider
the dihedral group D2·k , that is, the group of isometries of a regular k-gon
K of edge lengths equal to 2. The embedding f of the interval [−1, 1] as
any edge of K yields a quasi-isometry of OD2·1 [−1, 1] and OD2·k K, where
D2·1 ..= {1, −1} acts antipodally on [−1, 1]. In fact, these are all quasiisometric to O{1} [0, 1], as [0, 1] ≃ K/D2·k
Proof. For item (1), note that by Lemma 2.5, we have
dF (ty, ty ′ ) = min td(f y, y ′ ) + |f |,
f ∈F

and the quotient metric, by definition, equals
d([ty], [ty ′ ]) = min
td(f y, f y ′ ) = min td(f y, y ′ ).
′
f,f ∈F

f ∈F

The same estimates hold for item (2), that is, the quotient map is a quasiisometry with constants C = 1 and A = diam F . Indeed, the quotient map
(tY, dΓ ) → (tY /F, dΓ/F ) is obviously 1-Lipschitz and it suffices to show that it
decreases distances by at most diam F .
Let y, y ′ ∈ Y and let the distance dΓ/F ([ty], [ty ′ ]) be realised by sequences
n−1
(yi )ni=0 ⊆ Y and (si )i=1
⊆ S with [y0 ] = [y] and [yn ] = [y ′ ], that is:
′

dΓ/F ([ty], [ty ]) = td([y0 ], [y1 ]) +

n−1
X

1 + td([si yi ], [yi+1 ])

i=1

(see [24, Proposition 1.6]). (One should think about the sequence [y0 ], [y1 ], [s1 y1 ],
[y2 ], . . . , [sn−1 yn−1 ], [yn ] – then 1 comes from the definition of warped metric as an
upper bound for dΓ/F ([tyi ], si [tyi ]) and similarly td([si yi ], [yi+1 ]) is an upper bound
for dΓ/F ([si tyi ], [tyi+1 ]).)
We can assume that y0 = y, then that d(y0 , y1 ) = d([y0 ], [y1 ]), and further, by an
inductive argument, that td(si yi , yi+1 ) = td([si yi ], [yi+1 ]). It may happen that yn 6=
y ′ , but we still have dΓ (yn , y ′ ) ≤ diam F (as [yn ] = [y ′ ]). Consequently, by applying
the definition of warped metric and the triangle inequality to sequence y0 , y1 , s1 y1 ,
y2 , . . ., sn−1 yn−1 , yn , y ′ , we conclude that dΓ (ty, ty ′ ) ≤ dΓ/F ([ty], [ty ′ ]) + diam F .

Items (3) and (4) follow from item (1).
5. Dependence on metric
We already know that quasi-isometry of warped cones does not imply quasiisometry of groups or a homeomorphism of the spaces acted upon, but if the the
quasi-isometry is induced by a continuous map f , then the groups must be virtually
isomorphic and the virtual isomorphism commutes with f (Theorem 4.1). One can
ask whether the converse holds.
Question 5.1. If two actions are similar, are the respective warped cones quasiisometric?
This also turns out to be false, in the strongest possible sense.
Proposition 5.2. There exists an action ΓyY and two invariant metrics d, d′ on
the topological space Y such that the warped cones OΓ (Y, d) and OΓ (Y, d′ ) are not
quasi-isometric.

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DAMIAN SAWICKI

The above proposition is quite trivial, the way that it is stated – one can consider
the trivial action on some Y with some natural metric d′ and find a sufficiently “wild”
metric d.
What we will actually prove is that there is a subgroup action ΓyY and invariant
metrics d, d′ such that the open cones O(Y, d) and O(Y, d′ ) have both asymptotic
dimension 0 (in particular, they have property A and embed coarsely into a Hilbert
space, so neither d nor d′ should be considered “wild”), moreover there is an isometric
embedding (ft : (tY, d) → (tY, d′ ))t>0 , but there is no coarse embedding OΓ (Y, d) →
OΓ (Y, d′ ), because OΓ (Y, d) does not admit a coarse embedding into a Hilbert space
and OΓ (Y, d′ ) does. In particular, OΓ (Y, d) and OΓ (Y, d′ ) are not-quasi isometric.
In other words, we cannot predict properties (at least coarse embeddability) of
the warped cone by just looking at the group, action or the metric space Y , but
it is necessary to consider all these jointly. The difference between OΓ (Y, d) and
OΓ (Y, d′ ) is a purely metric phenomena coming from the interplay – in the definition
of warped metrics – of the action and metrics d, d′ (both very similar, if we forget
the action).
Theorem 5.3. There exists a compact space Y with an action of the free group F3
admitting two invariant metrics d and d′ such that
• both infinite cones O(Y, d) and O(Y, d′ ) have asymptotic Assouad–Nagata
dimension equal to 0;
• there is an isometric embedding f : (Y, d) → (Y, d′ );
• both warped cones OF3 (Y, d), OF3 (Y, d′ ) do not have property A;
• both warped cones OF3 (Y, d), OF3 (Y, d′ ) admit a fibred coarse embedding
into a Hilbert space;
however
• the warped cone OF3 (Y, d) does not admit a coarse embedding into the
Hilbert space and OF3 (Y, d′ ) does;
• in particular, there is no coarse embedding OF3 (Y, d) → OF3 (Y, d′ ), so they
cannot be quasi-isometric.
Moreover, the action is profinite, minimal, free and admits an ergodic invariant
probability measure (it is an action of a dense subgroup on an ambient compact
group).
Our construction relies on the examples of box spaces obtained recently by Delabie and Khukhro [9]. They constructed a sequence of normal subgroups in the
free group F3 E M1 E N1 E M2 E N2 E · · · with trivial intersection such that
the box space associated to sequence Mi does not embed coarsely into the Hilbert
space and the box space associated to Ni does.
To a chain of quotients Gi = Γ/Γi as in the definition of a box space (Subsection
b i )) = lim Gi ,
Examples in Section 2), one can also associate a compact group Γ((Γ
←−
called a profinite completion or the boundary of the coset tree. As Γ maps into
each Gi , it maps into the inverse limit, and this map is an injection onto a dense
b
subgroup. In particular, Γ acts
Q on Γ((Γi )) by left translations. The inverse limit
can be realised as a subset of i Gi
{(gi )i≥0 | gi−1 = q(gi ) ∀i ≥ 1}

(where q : Γ/Γi → Γ/Γi−1 is the quotient map) and hence it inherits the ℓ∞ -product
metric d((gi ), (hi )) = aj , where j is the first index with gj 6= hj and the sequence
aj > 0 converges monotonically to 0.
The author showed in [27] that the box space and the completion as above are
b i )) then it has property
tightly related. If one constructs the warped cone over Γ((Γ
A if and only if the box space (Gi )i does. Moreover, if the rate of convergence of aj

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

13

is large enough, then the warped cone embeds coarsely into the Hilbert space if and
only if the box space does. Furthermore, the box space embeds quasi-isometrically
into the warped cone. However, it was not clear how important the choice of metric
is to these results. Theorem 5.3 shows that it is indeed crucial.
Note also that during his lecture in Cambridge in February 2017 Pierre Pansu
asked the following (see https://metric2011.wordpress.com/2017/02/07/ for
the exact statement):
b i )) quasi isometric to the box
Question 5.4. (Pansu) Is the warped cone over Γ((Γ
space (Γ/Γi )i ?

Theorem 5.3 provides a rather negative answer to this question – while by [27]
one may construct a coarse (in fact an almost isometric) embedding (Γ/Γi ) →
b i )), this is conditional on a suitable choice of metric on Γ((Γ
b i )) and the
OΓ Γ((Γ
b
coarse geometry of OΓ Γ((Γi )) depends on this choice, in particular it is not always
the same as the geometry of the box space (Γ/Γi ).

Definition 5.5. For a subset U of some metric space and R > 0, an R-component
of U is a class of the equivalence relation on U spanned by identifying points at
distance smaller than R.
For d ∈ N, a metric space X has asymptotic dimension at most d, denoted
asdim X ≤ d, if for every number R ∈ N+ there exists another number S(n) ∈ N
and a covering U0 , . . . Ud of X such that diameters of R-components of sets Ui are
bounded by S. Minimal such d is called the asymptotic dimension of X (if there is
no d as above, we say that the dimension is infinite).
If one requires S to be a linear function of R, one obtains the definition of
asymptotic Assouad–Nagata dimension and if additionally R ranges over (0, ∞)
rather than N+ – Assouad–Nagata dimension, which are denoted by asdimAN and
dimAN , respectively.

Proof of Theorem 5.3. Consider a sequence of finite index normal subgroups in the
free group F3 E M1 E N1 E M2 E N2 E · · · such that the box space Gi = F3 /Mi
does not embed coarsely into the Hilbert space and the box space Hi = F3 /Ni does,
i
i
as obtained in [9]. After passing to a subsequence we can assume GGi−1
≤ HHi−1
. We
also assume that the sequence (ai ) decreases sufficiently fast for the results of [27]
to apply (to both box spaces (Gi ) and (Hi )).
Clearly, the inverse limit Y of the inverse system G1 ← H1 ← G2 ← H2 ← · · ·
is the same as the inverse limit of the subsystem of Gi ’s or the Q
subsystem of Hi ’s.
Let d denote the metric on Y induced from the embedding
into
Gi and d′ denote
Q
the metric on Y induced from the embedding into Hi . We denote the images of
these embeddings by G and H.
Asymptotic dimension. We will first verify that (tY, d)t>0 has the asymptotic
(Assouad–Nagata) dimension 0 (the proof for (tY, d′ )t>0 is literally the same). Let
R ∈ N+ and t > 0. Since we want to prove that the dimension is zero, it must
be U0 = tY . Let j ∈ N be the smallest number such that taj < R. Then an
R-component of tY consists of points t(gi ) with prescribed values of all coordinates
gi with i < j. Indeed, if a point t(b
gi ) lies outside such a set, then, for any t(gi )
from our set we have gbi 6= gi for some i < j and hence d(t(b
gi ), t(gi )) ≥ tai ≥ R.
Also, two points from an R-component can differ only on coordinates of index at
least j, so the diameter of an R-component is taj < R, and we can put S = R.
Isometric embedding. Now, we will construct an isometric embedding e : (Y, d) →
(Y, d′ ). We will define injective maps ei : Gi → Hi such that q ◦ ei = Q
ei−1 ◦ q,
Qwhere
q’s
are
the
appropriate
quotient
maps,
and
hence
the
product
map
e
:
Gi →
i
Q
Hi will map G into H.

14

DAMIAN SAWICKI

Let e1 be any injective map from G1 to H1 (we assume G0 = H0 = {∗} so our
i
i
assumption GGi−1
≤ HHi−1
allows us to do that). Then, e2 must map any element

G2
2
g2 ∈ G2 into the set q −1 (e1 (q(g2 ))). This set has H
H1 elements and there are G1
elements in the set q −1 (q(g2 )) that must be mapped there. The existence of e2
follows from our assumption on cardinalities and the existence of e – by induction.
Property A and fibred embeddability. Since F3 is non-amenable, lack of property
A follows from [24, Proposition 4.1] (it is formulated in a less general setting, but
the proof remains valid in our case; alternatively, one can use [27, Proposition 6.1]).
The existence of a fibred coarse embedding was proved in [30, Theorem 3.2 together
with Lemma 3.14].
(Non-)embeddability. Embeddability of a warped cone coming from an embeddable box space follows from [27, Theorem 7.4] and the analogous statement for
non-embeddability follows from [27, Corollary 7.6].


6. Asymptotically faithful coverings and piecewise properties
In this section, we introduce piecewise versions of metric invariants, which – for
spaces X which cover asymptotically faithfully a sequence of spaces Yi – enables
us to characterise properties of X by the piecewise versions of these properties for
(Yi ). The bulk of this section is devoted to proving simple lemmas establishing
properties of this new notion.
Inspiration comes from the following recent question of Osajda.
Question 6.1 (Osajda [20]). What are coarse geometric properties of box spaces
of groups without property A? Can lack of property A of a group be characterized
by coarse geometric properties of its box space?
Osajda explains in [20] that the answer to his question is affirmative by unpublished results of Thibault Pillon. Our result is more general than Osajda’s question,
apart from property A, it applies for instance to hyperbolicity, asymptotic dimension or coarse embeddability into Banach spaces. In particular, Pillon calls his
notion fibred property A, which resembles fibred coarse embeddability of [8], but
our piecewise coarse embeddability is much weaker than fibred coarse embeddability.
While it is theoretically appealing that we can, in particular, characterise many
geometric properties of groups by geometric properties of their box spaces (so far,
we could do that for “group-theoretic” or “equivariant” properties like amenability
[23] or the Haagerup property [7, 8, 23] (more generally, property P B [1])), we are
mostly interested in applications of this new notion. These are provided, in the case
of asymptotic dimension, by a recent result of Yamauchi [37], which we formulate
at the end of this section using the general language developed here. Applications
to warped cones are given in the next section.
In what follows, we say that a collection of spaces, say a warped cone (tY, dΓ )t>0 ,
has some property P if all the spaces in the collection satisfy property P with
a uniform estimate on the respective constants. Examples of properties include
hyperbolicity (with a uniform constant δ), coarse embeddability into a Banach space
E (with uniform estimates ρ− , ρ+ as in Definition 2.2), asymptotic dimension at
most k (with a uniform function R 7→ S(R)) or property A (with a uniform function
R 7→ S(R), see Definition 6.7).
This is in accordance with the language we have been using so far, when we
called a family of quasi-isometric maps with uniform constants a quasi-isometry.
Such an approach was introduced in order to define finite decomposition complexity
in [12], where such collections are called metric families. This is mainly a matter of
a convenient language, as, in particular, for the properties that we will be interested

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

in, a property holds for a box space
family (Γ/Γi )i .

F

i

15

Γ/Γi if and only if it holds for for the metric

Definition 6.2. Let P be some property of metric spaces (that is, invariant under
isometries).
• A family of metric spaces (Yi )i∈I indexed by a linearly ordered set (I, ≤)
has property
P piecewise if for every R ∈ N there is j ∈ I such that the
S
R
sum R Yj(R)+
of metric families
R
Yj+
= {A | ∃i ≥ j : A ⊆ Yi and diam A ≤ R}

satisfies property P . (We will only consider the case I = N and I = (0, ∞).)
• Respectively, a metric space Y has property P piecewise
S if for every R ∈ N
there is a bounded subset F ⊆ Y such that the sum R YFR(R)+ of metric
families
YFR+ = {A | A ⊆ Y \ F and diam A ≤ R}

satisfies property P .
• Property P is inherited by subsets if for any metric family Y having property
P , also the family Y⊇ = {A | ∃Y ∈ Y : A ⊆ Y } has property P .
• Property P is finitely determined if for any proper and discrete metric space
X, the following implication holds: if the metric family
X<∞ = {A ⊆ X | diam A < ∞}
satisfies property P , then X satisfies property P .
• Property P is coarsely invariant if for any metric families (Yi ) and (Xi )
and a coarse equivalence (fi : Yi → Xi ) the family (Yi ) satisfies property P
if and only if (Xi ) does.
Clearly, property P implies piecewise property P under the assumption that P
is inherited by subsets.
It is well-known that properties like property A, asymptotic dimension at most
k or coarse embeddability into some Banach space E are all inherited by subsets
and coarsely invariant. Hyperbolicity (with a fixed constant δ) is obviously finitely
determined, as it can be defined using Gromov’s “four point condition”. The fact
that coarse embeddability into a fixed Banach space is finitely determined is the
main result of [21]. We verify finite determination for property A and asymptotic
dimension in Proposition 6.8.
Let us check some basic properties of our definitions.
Lemma 6.3. If P is coarsely invariant and (fi : Yi → Xi ) is a coarse equivalence,
then (Yi ) has property P piecewise if and only (Xi ) does.
Proof. It suffices to show that if (Xi ) has property P piecewise, then (Yi ) also does.
So assume that (Xi ) has property P piecewise.
R
R
Let R < ∞. For any j ∈ I we can consider the “image” of Yj+
, that is, f (Yj+
)=
{fi (A) | i ≥ j, A ⊆ Yi and diam A ≤ R}. Since (fi ) is a coarse equivalence, there
R
exists S(R) < ∞ (not depending on j) such that diam B < S(R) for all B ∈ f (Yj+
).
Now, we know that there is j(S(R)) such that
[ S(R)
Xj(S(R))+
R∈N

S
R
).
satisfies property P and hence the same holds for its subfamily R∈N f (Yj(S(R))+
S
R
Since P is a coarse invariant, R∈N Yj(S(R))+ satisfies property P as well, which
finishes the proof.


16

DAMIAN SAWICKI

Clearly, the same can be said about quasi-isometrically invariant properties and
quasi-isometric maps and alike for rough isometries (that is, quasi-isometries with
multiplicative constant C = 1), also known as almost isometries.
Let us verify that piecewise properties for disjoint sums and sequences are equivalent (even if P itself is not a coarse invariant).
Lemma 6.4. Let (Yi ) be a sequence of bounded metric spaces and P be any property.
Then the F
sequence (Yi ) has piecewise property P if and only if the coarse disjoint
sum Y = i Yi has piecewise property P .

Proof. Let us assume that (Yi ) has piecewise
P , that is, for every R ∈
S property
R
satisfies property P . Let
N there is j(R) ∈ N such that the sum R Yj(R)+
J(R) ≥ j(R) be so large that d(Yk , Yk′ ) > R inside Y for k, k ′ ≥ J(R). Then,
SJ(R)−1
R
for F (R) = i=1
Yi the family YFR(R)+ is a subfamily of Yj(R)+
, and hence the
S
sum R YFR(R)+ has property P .
Sj(R)−1
In the other direction, it suffices to take j(R) so large that F (R) ⊆ i=1
Yi
R
and then the family Yj(R)+
is a subfamily of YFR(R)+ .

The main motivation for introduction of piecewise properties comes from the
following two lemmas that lead to characterisations (see Corollary 6.9 and Proposition 7.4) and, when combined with other results [37], to some unexpected equalities
(see Corollary 6.11 and Theorem 7.1).
Lemma 6.5. If fi : Xi → Yi is an asymptotically faithful covering and X = (Xi )
satisfies property P which is inherited by subsets, then (Yi ) has property P piecewise.
Proof. Let R ∈ N. By asymptotic faithfulness there exists j(R) ∈ I such that the
R
inverse image of any A ∈ Yj(R)+
via the appropriate fi consists of its isometric
S
R
.
copies. Consider the family U .= R∈N f −1 (Yj(R)+
) consisting of such copies of all
S
R
A ∈ R∈N Yj(R)+ =.. D. Since U is a subfamily of X⊇ and property P is inherited
R
by subsets, family U satisfies P . As it consists of copies of elements of Yj+
, the
same is holds for family D.

Lemma 6.6. Let X be a discrete proper metric space, P be finitely determined
and fi : X → Yi be an asymptotically faithful covering such that (Yi ) has property
P piecewise. Then X has property P .
Proof. Let R ∈ N and let A ⊆ X have diameter bounded by R. By the asymptotic
faithfulness there is j(R) ∈ N such that fi : B(x, R) → B(fi (x), R) is an isometry
for any x ∈ X and i ≥ j. By piecewise property P we may increase j(R) so that
the family
{fi (A) | R ∈ N, i ≥ j(R), A ⊆ X and diam A ≤ R}
S
R
) satisfies property P . But the above family is isometric
(as a subfamily of R Yj(R)+
to
{A | A ⊆ X and diam A < ∞} = X<∞
and hence X has property P by finite determination.

We will use the following definition of property A (cf. [30]), equivalent to the
original one [38] for bounded geometry metric spaces [24].
Definition 6.7. A metric space X has property A for every R ∈ N there is S ∈ N
and a map f : X → Prob(X) such that kf (x) − f (x′ )k ≤ 1/R for d(x, x′ ) ≤ R and
supp f (x) ⊆ B(x, S).
Proposition 6.8. Property A and having asymptotic dimension at most k are
finitely determined.

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

17

Proof. Let X be a discrete proper metric space.
Assume first that X<∞ has asymptotic dimension at most k. That is, for any R ∈
N there is S ∈ N such that for any A ⊆ X of finite diameter we have sets U0A , . . . , UkA
Sk
such that i=0 UkA = A and R-components of Ui have diameter bounded by S. We
can assume that sets UiA are mutually disjoint, and hence the above sets UiA can
be encoded by a function fA : A → {0, . . . , k}.
Now, consider any non-principal ultrafilter U on N and let f : X → {0, . . . , k} be
the limit f (x) = limn→U fB(z,n) (x) for some fixed z ∈ X. Consider an R-component
of f −1 (i). If its diameter is not S-bounded, there must be a pair of points x, x′ ∈
f −1 (i) with d(x, x′ ) > S and a finite sequence x = x0 , x1 , . . . xm = x′ ∈ f −1 (i) with
d(xi−1 , xi ) < R. By properties of ultrafilters, there must be (infinitely many) n
B(z,n)
contains this sequence, and hence some R-component of Ui has
such that Ui
diameter more than S, a contradiction.
As for property A, assume that for every R ∈ N there is S ∈ N such that
for any A ⊆ X of finite diameter we have maps fA : A → Prob(A) such that
kfA (x) − fA (x′ )k ≤ 1/2R for d(x, x′ ) ≤ R and supp fA (x) ⊆ B(x, S).
Fix some R together with the respective constant S and functions fA . Since
B(x, S) is finite, say of size K, we can modify fA (x) to fA′ (x) ∈ Prob(B(x, S)) so
that measures of singletons with respect to fA′ (x) take values in the finite set


1
4KR − 1
0
,
,...,
,1
4KR 4KR
4KR

and kfA (x) − fA′ (x)k ≤ 1/4R, and hence we still know that kfA′ (x) − fA′ (x′ )k ≤ 1/R
for d(x, x′ ) ≤ R. What we have gained is that now the function A 7→ fA′ (x)
can acquire only finitely many values, so we can apply the previous trick – define
′
f : X → Prob(X) by f (x) = limn→U fB(z,n)
(x). By definition f (x) ∈ Prob(B(x, S))
′
′
and for any x, x with d(x, x ) ≤ R there is n such that
′
′
(x′ )k ≤ 1/R.
(x) − fB(z,n)
kf (x) − f (x′ )k = kfB(z,n)



Hence, we have obtained the following characterisation of metric invariants of a
group by metric invariants of its box space.
Corollary 6.9. Let P be “hyperbolicity”, “property A”, “asymptotic dimension at
most k” or “coarse embeddability into a Banach space E”. Let Γ be a residually
finite group with a box space (Γ/Γi ). Then, Γ has property P if and only if the box
space (Γ/Γi ) has property P piecewise.
The following surprising result was recently obtained by Takamitsu Yamauchi
[37]. It was formulated for large girth graphs and without using the language
of piecewise properties, but Yamauchi’s proof remains valid in this more general
setting.
Theorem 6.10 (Yamauchi [37]). Assume that a family of bounded spaces (Xi )i∈N
has piecewise asymptotic dimension at most d and that asdim(Xi )i ≤ n for some
n < ∞. Then in fact asdim(Xi )i ≤ d.
Proof. By the assumption, for every m ∈ N there is Sm ∈ N such that every Xi
m
m
admits a partition into sets Ui,0
, . . . , Ui,n
, whose m-components are Sm -bounded.
Additionally, for every S ∈ N there is jS ∈ N such that the family of sets XjSS + has
asymptotic dimension at most d. We can assume that m 7→ Sm and S 7→ jS are
strictly increasing.
Let us decompose every set Xi into n + 1 subsets Vi,0 , . . . , Vi,n as follows. For
m
. Hence, for
all i < jS1 we put Vi,l = Xi . For jSm ≤ i < jSm+1 we put Vi,l = Ui,l
jSm ≤ i < jSm+1 , m-components of Vi,l are Sm -bounded.

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DAMIAN SAWICKI

We will show that for every l ∈ {0, . . . , n} the family (Vi,l )i has asymptotic
dimension at most d. Fix one such l and let R ∈ N. For i < jSR we can dep
0
d
compose Vi,l as Wi,l
= . . . = Wi,l
= Vi,l and clearly R-components of Wi,l
are
(maxi<jSR diam(Vi,l ))-bounded. For m ≥ R and jSm ≤ i < jSm+1 , every mm
component of Vi,l = Ui,l
is Sm -bounded and in particular every R-component A
of Vi,l is Sm -bounded. Since i ≥ jSm , by the assumption of piecewise asymptotic
dimension at most d, every such component A admits a decomposition into sets
WA0 , . . . , WAd , whose R-components
S are T -bounded (where T depends on R, but
p
not on A or m). We put Wi,l
= A WAp – since the sum is over R-components of
Vi,l and WAp ⊆ A, we conclude that
Sn R-components of the sum are still T -bounded.
Since every Xi is a finite sum l=0 Vi,l (of the fixed length n + 1) and for every l
the family (Vi,l )i has asymptotic dimension at most d, by using the finite union theorem for asymptotic dimension [5] we conclude that the family (Xi ) has asymptotic
dimension at most d.

The following corollary was mentioned by Rufus Willett during his lectures in
Southampton in March 2017.
Corollary 6.11 (Yamauchi, Willett). Let (Γ/Γi ) be a box space. Then asdim(Γ/Γi )
is infinite or equals asdim Γ.
Proof. If asdim(Γ/Γi ) = ∞, there is nothing to prove, so assume that asdim(Γ/Γi )
is finite. Let k ≤ asdim(Γ/Γi ) be the minimal number such that (Γ/Γi ) has asymptotic dimension at most k piecewise. By Corollary 6.9 asdim Γ = k. But now, by

Theorem 6.10, we have asdim(Γ/Γi ) ≤ k.
7. Piecewise properties and warped cones
In this section we will apply the definitions and results of the previous section
to warped cones. The main result is as follows.
Theorem 7.1. Let M be an m-dimensional Riemannian manifold and ΓyM be
free and isometric. Assume that asdim OΓ M < ∞. Then
asdim OΓ M = asdim Γ × Zm .

Let Y be an ultrametric space with a Lipschitz action ΓyY and assume that
asdim OΓ Y < ∞, then
asdim OΓ Y = asdim Γ.
Before proving this, let us consider the following two propositions that gather
results on piecewise properties of warped cones.
Proposition 7.2. Let ΓyY be a Lipschitz action.
(1) (Γ × OY, d1 ) has property A if and only if both Γ and OY have property A.
(2) (Γ × OY, d1 ) has finite asymptotic dimension if and only if both Γ and OY
have finite asymptotic dimension; in fact,
asdim Γ, asdim OY ≤ asdim(Γ × OY, d1 ) ≤ asdim Γ + asdim OY.
(3) Assume that the action is isometric and E is a Banach space such that
E ⊕ E is isomorphic to a subspace of E. Then, clearly, (Γ × OY, d1 ) embeds
coarsely into E if and only if both Γ and OY do.

If the action is additionally free, and the equivalent conditions mentioned in (1),
(2) or (3) hold, then the warped cone OΓ Y has, respectively, piecewise property A,
piecewise finite asymptotic dimension of piecewise embeds coarsely into E.

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

19

Proof. For the “only if” part of (1), (2) and (3) it suffices to observe that Γ and
OY embed coarsely into (Γ × OY, d1 ). For Γ, the map γ 7→ (γ, x) is an isometric
embedding for any x ∈ OY . The case of OY is less obvious, but a reasoning similar
to that in in the proof of Lemma 2.5 shows that
(9)

′

d1 ((e, x), (e, x′ )) ≤ d(x, x′ ) ≤ Ld1 ((e,x),(e,x )) · d1 ((e, x), (e, x′ ))

where L is the Lipschitz constant for the action of generators, so indeed OY ∋ x 7→
(e, x) ∈ (Γ × OY, d1 ) is a coarse-embedding.
The “if” part of assertion (1) is proved in [28] and the “if” part of assertion (3)
is straightforward – it sufices to define the coarse embedding f : Γ × OY → E as
the sum of coarse embeddings fΓ : Γ → E and fOY : OY → E, namely
f (γ, x) = fΓ (γ) ⊕ fOY (x) ∈ E ⊕ E ⊆ E
(recall that when the action is isometric, then d1 is the product metric).
To verify the “if” part of assertion (2), we will use the following Hurewicz-type
theorem:
Theorem 7.3 (Brodskiy–Dydak–Levin–Mitra [6]). Let f : X → Z be a large-scale
uniform map. Then asdim X ≤ asdim Z + asdim f , where
asdim f = sup{asdim f −1 (A) | A ⊆ Z and asdim A = 0}.

Here, large-scale uniform (or bornologous in the terminology of J. Roe [23])
means that there exists a non-decreasing function ρ+ such that d(f (x), f (x′ )) ≤
ρ+ ◦ d(x, x′ ).
We define f : (Γ × OY, d1 ) → Γ as the projection onto the first coordinate.
Clearly, this is a 1-Lipschitz map. Let A ⊆ Γ have asymptotic dimension equal
to 0. That is, for every R ∈ N the diameter of R-components of A is uniformly
bounded by some constant D. By definition we have f −1 (A) = A×OY , in particular
for two different R-components C, C ′ ⊆ A the distance of f −1 (C) and f −1 (C ′ ) is
at least R. Consequently, in order to show that f −1 (A) can be partitioned into
d + 1 = asdim OY + 1 sets, whose R-components are uniformly bounded, it suffices
to partition every f −1 (C).
Hence, fix one R-component C of A and pick γ ∈ C. Using the isometric group
action on Γ × OY and the compatible action on Γ by right multiplication, we
can assume γ = e. Let U0 , . . . Ud be a partition of OY whose (LR+2D (R + 2D))components are S-bounded. Define W0 , . . . Wd by Wi = C × Ui .
Now if d1 ((η, x), (δ, z)) < R for η, δ ∈ C then we have d1 ((e, x), (e, z)) < R + 2D
and consequently, by (9), d(x, z) < LR+2D (R + 2D). Hence, we conclude that the
projection onto OY (given by (η, x) 7→ x) of an R-component of Wi is S-bounded.
Consequently, any such component must be (S + 2D) bounded. This proves that
asdim(Γ × OY, d1 ) ≤ asdim Γ + asdim OY .

The last part of the Proposition is a direct application of Lemma 6.5.
In particular, for reasonable spaces Y and free actions, we have the following
equivalences.
Proposition 7.4. Assume that OY has finite asymptotic dimension (which is the
case if Y is a manifold, finite simplicial complex or a profinite completion) and the
action ΓyY is free and Lipschitz. Then:
(1) Γ has property A if and only if OΓ Y has property A piecewise.
(2) Γ has finite asymptotic dimension if and only if OΓ Y has finite asymptotic
dimension piecewise.
(3) Assume that the action is isometric. Then, Γ embeds coarsely into E if and
only if OΓ Y embeds coarsely into E piecewise.

20

DAMIAN SAWICKI

Where E is a Banach space such that there exists an infinite dimensional space B
such that E ⊕ B is isomorphic to a subspace of E.
In fact, only one free orbit is used in the proof of the “if” parts of assertions (1),
(2), and (3).
Proof. The “only if” parts follows from Proposition 7.2 and the fact that finite asymptotic dimension implies property A, which implies coarse embeddability into
a Hilbert space [38]. Since levels of the warped cone are compact spaces, we can
embed them into finite-dimensional Hilbert spaces as well (with the dimension depending on the level!). As the family of Euclidean spaces (Ri )i embeds bi-Lipschitz
into any infinite dimensional Banach space B, we get that OY embeds coarsely into
B.
For the “if” part recall from Lemma 2.6 that for y0 ∈ Y belonging to a free orbit,
we have lim d(ty0 , γty0 ) = |γ| and in fact the function t 7→ d(ty0 , γty0 ) is constant
for large t. Hence, every finite subset A of Γ can be isometrically realised as a
subset Aty0 ⊆ tY for sufficiently large t. By further increasing t we may assume
that t ≥ j(diam A), where j(diam A) is the appropriate constant from the definition of piecewise property A, piecewise asymptotic dimension or piecewise coarse
embeddability into E. Since the considered properties are finitely determined, this
ends the proof.

By combining Theorem 6.10 with Proposition 7.2 and Theorem 2.4, one obtains:
Corollary 7.5. Assume that Γ and OY have finite asymptotic dimension and the
action ΓyY is Lipschitz and free. Then, if asdim OΓ Y < ∞ then in fact
asdim OΓ Y ≤ asdim Γ + asdim OY.
one can prove that the unified warped cone OΓu Y =
S Note that using Theorem 7.3
u
t>0 (tY, dΓ ) satisfies asdim OΓ Y ≤ asdim OΓ Y + 1.
We will now proceed to the proof of Theorem 7.1, which is a sharper version of
the above Corollary 7.5 valid for free isometric actions on manifolds.
Proof of Theorem 7.1. Note that it follows from our assumptions and the proof of
Theorem 3.1 that for every R ∈ N there is t0 such that for t > t0 any R-ball in
(tM, dΓ ) is quasi-isometric (with constants not depending on R) to an R-ball in
Γ × Rm (in fact one gets a Lipschitz equivalence and the Lipschitz constant can
be taken arbitrarily close to 1, but we do not need it here or in Theorem 3.1),
and further to an R-ball in Γ × Zm . Consequently, the warped cone has piecewise
asymptotic dimension at most asdim Γ×Zn . Then, by the theorem of Yamauchi [37]
(Theorem 6.10 above), we have asdim OΓ M ≤ asdim Γ × Zn . But this inequality
cannot be strict – this would imply that the asymptotic dimension of the family of
balls in Γ × Zn is strictly smaller than the asymptotic dimension of Γ × Zn , which
is impossible as asymptotic dimension is finitely determined by Lemma 6.8.
If (Y, d) is an ultrametric space, then for any R > 0 its R-components are
R-bounded, whence asdim OY = 0 just as in the proof of Theorem 5.3. By Corollary 7.5 one gets asdim OΓ Y ≤ asdim Γ. The opposite inequality follows from finite
determination of the asymptotic dimension (Proposition 6.8) and the fact that Γ
embeds isometrically into (Γ × OY, d1 ), and hence finite subsets of Γ embed isomet
rically into OΓ Y by Theorem 2.4.
8. Applications to dynamics and C ∗ -dynamics
The following was obtained by Jianchao Wu and Joachim Zacharias; we use the
standard notation in dimension theory, where dim+1 (·) means dim(·) + 1.

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

21

Theorem 8.1 (Wu–Zacharias, unpublished). For a free and Lipschitz action ΓyY ,
we have the following relations between equivariant asymptotic dimension of the
action and asymptotic dimension of the warped cone:
(1) asdim+1 (OΓ Y ) ≤ asdim+1 (OY ) · eq-asdim+1 (ΓyY );
(2) eq-asdim(ΓyY ) ≤ asdim(OΓ Y ),
where eq-asdim denotes the equivariant asymptotic dimension with respect to the
trivial family of subgroups of Γ, namely {{e}}.
In particular, if asdim OY < ∞, then
asdim OΓ Y < ∞ ⇐⇒ eq-asdim(ΓyY ) < ∞.

We do not provide the definition of the equivariant asymptotic dimension of the
action (with respect to a family of subgroups F of Γ, denoted F-eq-asdim(ΓyY )),
because all that we will use is the above theorem. Instead, we refer the reader to [26]
(see also [13, 32]). We remark that eq-asdim is also called amenability dimension
by some authors [32], especially when F consists only of the trivial subgroup {e}
as above. This notion was introduced as a technical condition (without an explicit
definition, hence different names used by different authors) in the proof of the
Farrell–Jones conjecture for hyperbolic groups by Bartels, Lück, and Reich [4].
Equivariant asymptotic dimension relates to many other properties like dynamic
asymptotic dimension [13] of group actions as well as Rokhlin dimension of C ∗ dynamical systems and nuclear dimension of C ∗ -algebras [32].
Recall the following result.
Theorem 8.2 (Szabo–Wu–Zacharias [32]). Let ΓyY be a free action of a nilpotent
group Γ on a metrisable space Y . Then:
eq-asdim+1 (ΓyY ) ≤ 3asdim Γ · dim+1 Y,

where dim Y is the Lebesgue covering dimension of Y .

The original result [32, Theorem 7.4 with Lemma 8.4] is formulated for the
Hirsch length instead of the asymptotic dimension, but they are equal for virtually polycyclic groups (in particular for nilpotent groups) by a classical result of
Dranishnikov and Smith [11].
The following more general result is a special case of [3, Corollary 1.10].
Theorem 8.3 (Bartels [3]). Let ΓyY be a free action of a virtually nilpotent group
Γ on a metrisable space Y of finite covering dimension. Then:
eq-asdim(ΓyY ) < ∞.
At a seminar lecture in Warsaw in June 2017 Willett posed the following problem.
Problem 8.4 (Willett). Improve these upper bounds on the equivariant asymptotic
dimension.
The following corollary addresses the problem, giving a common bound in the
nilpotent and virtually nilpotent case. For 0-dimensional topological systems, this
bound becomes an equality.
Corollary 8.5. Let ΓyY be a free action of a virtually nilpotent group Γ on a
compact metrisable Y . Then:
eq-asdim(ΓyY ) ≤ asdim Γ + dim Y.

In fact, if dim Y = 0 (for example if Y is a profinite completion), then
eq-asdim(ΓyY ) = asdim Γ.
We will need the following lemma.

22

DAMIAN SAWICKI

Lemma 8.6. Let ΓyY be an action of a finitely generated group on a compact
metrisable space Y . There exists a metric d on Y such that Γ acts on (Y, d) by Lipschitz homeomorphisms. Moreover, it can be arranged that dimAN (Y, d) = dim Y .
Proof. By results of Nagata [18] and Assouad [2] a compact metrisable space Y
has topological dimension at most m if and only if it admits a metric d0 such that
(Y, d0 ) has the Assouad–Nagata dimension at most m, hence we can take d0 such
that dimAN (Y, d0 ) = dim Y .
Let c0 be the maximum value acquired by d0 . We will construct a function
c : [0, c0 ] → [0, 1] such that d = c ◦ d0 is the required metric. Let us inductively
define an auxiliary sequence cn by the formula
cn+1 = min {cn /3, min {d0 (sy, sy ′ ) | s ∈ S, d0 (y, y ′ ) ≥ cn }} ,
where the inner minimum is well-defined and positive by compactness of Y . Define
c(cn ) = 2−n , c(0) = 0 and extend it in an affine way on intervals [cn+1 , cn ]. Note
that we have the following inequality of difference quotients:
0<

2−n−1
2−n−1 − 2−n−2
2−n−2
2−n−2
2−n − 2−n−1
≤
≤
,
=
≤
cn − cn+1
2cn /3
cn /3
cn+1
cn+1 − cn+2

and so c is a concave and increasing function. Thus, the composition d = c ◦ d0 is
still a metric, that is, it satisfies the triangle inequality (see e.g. [25]).
Note that the action of every generator s ∈ S is 4-Lipschitz with respect to d.
Indeed, let y 6= y ′ ∈ Y , s ∈ S and let n be such that d0 (sy, sy ′ ) ∈ [cn+1 , cn ]. Then,
because s−1 ∈ S, we have
d0 (y, y ′ ) = d0 (s−1 sy, s−1 sy ′ ) ≥ cn+2

and hence
d(sy, sy ′ ) ≤ 2−n = 4 · 2−n−2 ≤ 4d(y, y ′ ).

Our change of metric does not increase the Assouad–Nagata dimension (cf. [25]
for the case of asymptotic Assouad–Nagata dimension). Indeed, R-components of
subsets of Y with respect to d are the same as c−1 (R)-components with respect to
d0 . For metric d0 there exists C < ∞ such that for every R ∈ (0, c0 ] there exist
a covering U0 , · · · , Udim Y of Y such that c−1 (R)-components of Ui are C · c−1 (R)bounded. Since ci+1 /ci ≤ 1/3, if c−1 (R) ∈ [cn+1 , cn ], then C · c−1 (R) ≤ cn−⌈log3 C⌉
(formally, one needs to extend the sequence cn by putting cn = c0 · 3−n for n < 0;
the respective extension of the map c is given by putting c(r) = 1 for r ≥ c0 ).
Coming back to metric d, we have R = c(c−1 (R)) ≥ 2−n−1 and
c(C · c−1 (R)) ≤ c(cn−⌈log3 C⌉ ) ≤ 2−n+⌈log3 C⌉ ,

so R-components of Ui are (2⌈log3 C⌉+1 · R)-bounded.



Proof of Corollary 8.5. Consider a metric d from Lemma 8.6 such that dimAN (Y, d)
equals dim Y . By definition,
(10) asdim O(Y, d) ≤ asdimAN O(Y, d) ≤ dimAN O(Y, d) = dimAN (Y, d) = dim Y.
By Theorem 8.3 and the assumption that dim Y < ∞, also eq-asdim(ΓyY ) is
finite and hence, by Theorem 8.1 item (1) and by inequality (10), asdim(OΓ Y ) is
finite, too. By Corollary 7.5 it is at most asdim Γ + asdim OY ≤ asdim Γ + dim Y .
We conclude by Theorem 8.1 item (2).
For the second assertion it suffices to additionally observe that in Theorem 8.1,
if asdim OY = 0, one gets an equality eq-asdim(ΓyY ) = asdim OΓ Y , and in
this case we also have asdim OΓ Y = asdim Γ, which follows from the proof of
Theorem 7.1.


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23

Acknowledgements
The author is indebted to Damian Orlef for inspiring discussions on diophantine
approximations, in particular for suggesting a proof of Proposition A.3. The author is grateful to Jianchao Wu and Joachim Zacharias for permission to use their
unpublished result, Theorem 8.1.
The author was partially supported by Narodowe Centrum Nauki grant Preludium number 2015/19/N/ST1/03606.
Appendix A. Approximating the badly approximable
by Dawid Kielak and Damian Sawicki
In the appendix we prove the following results.
Theorem A.1. Let α ∈ (0, 1) and let Z act on the circle T1 = R/Z by shifts
by α. Then the warped cone (tT1 , dZ )t>0 is quasi-isometric with the collection of
tori (T2 , τ d)τ >0 if and only if α is a restricted irrational number (see Definition
A.6).
We also offer a result about actions of the infinite dihedral group
D∞ = Z/2Z ∗ Z/2Z = {1, r} ∗ {1, r′ }.

Corollary A.2. Let D∞ yT1 be such that r and r′ act by non-trivial reflections.
Then the warped cone (tT1 , dD∞ )t>0 is quasi-isometric with the collection of spheres
(S2 , τ d)τ >0 if and only if r′ r acts by a rotation by a restricted irrational number.
. . . , αm ) ∈
In fact, Theorem A.1 generalises to actions of Zm . For α = (α1 ,P
(0, 1)m we consider the action Zm yα T1 given by (n1 , . . . , nm ).z = z + ni αi .

Proposition A.3. Let m ∈ N+ . There exist strictly monotonic sequences (ti ) and
(τi ) in N and uncountably many m-tuples (α1 , . . . , αm ) such that the subcollection
(ti T1 , dZm )i of the warped cone over Zm yα T1 is quasi-isometric with the collection
of tori (Tm+1 , τi d)i .

For α = (α1 , . . . , αm ) ∈ (0, 1)m we define the action Zm yα Tm by the formula
(ni )i .(zi )i = (zi + ni αi )i . When m = 1, we get the same action as in Theorem A.1,
so there is no ambiguity.
Example A.4. There exist strictly monotonic sequences (ti ), (τi ), and (ri ) in N
and uncountably many triples β = (β1 , β2 , β3 ) ∈ (0, 1)3 such that for any pair of
restricted irrationals α = (α1 , α2 ) and actions Z3 yβ T1 and Z2 yα T2 we get quasiisometries
(ti T1 , dZ3 )i ≃ (τi T2 , dZ2 )i ≃ (T4 , ri d)i .

The reason for the above is that (τi T2 , dZ2 ) can be decomposed as a Cartesian
product (τi T, dZ )×(τi T, dZ ), where the Z-action on each factor is given by rotations
by α1 or α2 and each factor (τi T, dZ ) is quasi-isometric to (T2 , ri d) by Theorem A.1.
All of the results above show that there exist quasi-isometric warped cones (or
subcollections thereof) over actions of non-quasi-isometric groups on non-homeomorphic spaces (actions of infinite groups are also not conjugate in any sense to
actions of the trivial group). In particular, the Zm and Zn factors are necessary in
Theorem 3.1.
The above results rely on the following, more technical, result.
Lemma A.5. Let m ∈ N+ , (α1 , . . . , αm ) ∈ (0, 1)m and 0 < K < ∞. Consider a
pair (q, l) ∈ N+ ×[1, ∞) with q ≤ l and assume that there exist (pi ) ∈ {1, . . . , q−1}m
such that
• |qαi − pi | < K/l for all i ∈ {1, . . . , m}, and

24

DAMIAN SAWICKI

∼ L Z/li Z such that the
• the group Z/qZ admits a decomposition Z/qZ =
i
image of pi in Z/qZ under this isomorphism is a generator of Z/li Z.

my
1
1
Then, for the action ZQ
α T , the level (qlT , dZm ) of the warped cone is quasiisometric to (T1 , ld) × i (T1 , li d), with quasi-isometry constants depending only on
the dimension m and the quality of approximation K.
Q
Proof. Let us put t = lq and denote the ℓ1 -metric on Tm+1 = (T1 , ld) × i (T1 , li d)
by D. In order to construct a quasi-isometry (tT1 , dZm ) → (Tm+1 , D), we will use
an intermediate space (tT1 , d′Zm ), which is defined as tT1 with the metric warped
with respect to the action Zm yβ T1 by the m-tuple of angles β = (β1 , . . . , βm ) with
βi = pi /q.
First note that the identity map id : (tT1 , dZm ) → (tT1 , d′Zm ) is bi-Lipschitz.
Indeed, recall that for isometric actions ΓyY , we have

dΓ (ty, ty ′ ) = min td(y, γy ′ ) + |γ|
γ∈Γ

by Lemma 2.5, and so we get the following inequality (we suppress t in the notation
for points in tT1 as t = lq is fixed):

dZm (z, z ′ ) = min m td(z, (ni ).z ′ ) + |(ni )|
(ni )∈Z
X
X

ni αi ) +
|ni |
= min m td(z, z ′ +
(ni )∈Z
X
X
X

ni pi /qi ) + t
|ni | · |αi − pi /q| +
|ni |
≤ min m td(z, z ′ +
(ni )∈Z
X
X
X

ni pi /qi ) + lq
|ni | · |αi − pi /q| +
|ni |
= min m td(z, z ′ +
(ni )∈Z
X
X

ni pi /qi ) + (K + 1)
|ni |
≤ min m td(z, z ′ +
(ni )∈Z

≤ (K + 1)d′Zm (z, z ′ )

(the converse estimate has the same proof).
Q
Observe that, by the Chinese Remainder Theorem, pi = j6=i lj · p′i for some
p′i ∈ {1, Q
. . . , li − 1} which generates Z/li Z. Let ri′ ∈ {1, . . . , li − 1} be such that
′
ri = ri · j6=i lj is an inverse of p′i modulo li . Our quasi-isometry ι : (tT1 , d′Zm ) →
Q
(T1 , ld) × i (T1 , li d) is given by
ι(z) = (qz, r1 z, . . . , rm z).

We will first check that some metric neighbourhood (depending only on m) of the
image of ι is the whole of (Tm+1 , D). Note that qβi ≡ 0 (mod 1) and also ri βj ≡ 0
(mod 1) for i 6= j, because βj = p′j /lj and lj divides ri . Consequently, we have the
following formula
(11)

ι((ni ).z) = (qz, r1 z + n1 /l1 , . . . , rm z + nm /lm ).
Q
Thus, for any ζ ∈ (T1 , ld) × i (T1 , li d), we can find z ∈ T1 and (ni ) such that
ι((ni ).z) has the first coordinate equal to the first coordinate of ζ and the remaining
coordinates of ι((ni ).z) and ζ differ by distances at most 1/2.
It remains to check that ι is a bi-Lipschitz map onto its image. Let z, z ′ ∈
(tT1 , d′Zm ) and let (ni ) ∈ Zm be such that d′Zm (z, z ′ ) = td(z, (ni ).z ′ ) + |(ni )|. Then
by the triangle inequality we have

D ι(z), ι(z ′ ) ≤ D(ι(z), ι((ni ).z ′ )) + D(ι((ni ).z ′ ), ι(z ′ ))

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

25

and the first summand can be bounded above as follows
X
lj d(rj z, rj · (ni ).z ′ )
D(ι(z), ι((ni ).z ′ )) = ld(qz, q · (ni ).z ′ ) +
≤ (lq +

X

j

lj rj )d(z, (ni ).z ′ ) ≤ (m + 1)td(z, (ni ).z ′ ),

as rj , lj ≤ q ≤ l and t = ql. As for the second summand, using formula (11), one
immediately obtains D(ι((ni ).z ′ ), ι(z ′ )) ≤ |(ni )|. We have just obtained that ι is
Lipschitz:
D(ι(z), ι(z ′ )) ≤ (m + 1)td(z, (ni ).z ′ ) + |(ni )| ≤ (m + 1)d′Zm (z, z ′ ).

Note that ι is a group homomorphism and all metrics are translation invariant. Hence, when proving the converse estimate, we can assume z ′ = 0. Let
θ ∈ (−1/2q, 1/2q] and ni ∈ (−li /2, li /2] ∩ Z be (the unique numbers) such that
X
z≡
ni βi + θ (mod 1).
We have

X

mi βi ) + |(mi )|
min m td(z,
(mi )∈Z
X
ni βi ) + |(ni )| = t|θ| + |(ni )|,
≤ td(z,

d′Zm (z, 0) =

and also

D(ι(z), ι(0)) = ld(qθ, 0) +

X

li d(ni /li + ri θ, 0).

i

If 2t|θ| ≥ maxi |ni | ≥ |(ni )|/m, then we can bound:

D(ι(z), ι(0)) ≥ ld(qθ, 0) = lq|θ| ≥ d′Zm (z, 0)/(2m + 1)

(where the equality follows from the assumption that θ ∈ (−1/2q, 1/2q]); otherwise,
for i such that ni = maxj nj , we have:
D(ι(z), ι(0)) ≥ li d(ri z, 0) = li d(ni /li + ri θ, 0)

≥ li d(ni /li , 0) − li d(ri θ, 0) = |ni | − li ri |θ|

≥ |ni | − t|θ| > |ni |/2 ≥ d′Zm (z, 0)/(2m + 1)
(where the second equality again uses assumptions on θ and on ni ). We have just
proved
(2m + 1)d′Zm (z, z ′ ) ≥ D(ι(z), ι(z ′ )) ≥ d′Zm (z, z ′ )/(2m + 1).

We will now deduce results A.1 – A.3 from Lemma A.5. For that, we need a bit
of number theory.
Given a sequence of integers (a0 , a1 , . . .) with ai > 0 for i > 0 one can form a
continued fraction
1
.
[a0 , a1 , . . .] = a0 +
a1 + a + 1 1
2

a3 + 1
···

The value of a continued fraction is the limit of the sequence of convergents:
1
1
1
a0 , a0 + , a0 +
, ...
, a0 +
a1
a1 + a12
a1 + a +1 1
2

a3

This establishes a bijection between irrational numbers and continued fractions. Let
us denote by pi ∈ Z and qi ∈ N+ the integers such that pi /qi is the ith convergent
and qi is minimal. In particular p0 = a0 and q0 = 1.
Definition A.6. An irrational number α is called restricted if the sequence ai is
bounded for the representation α = [a0 , a1 , . . .].

26

DAMIAN SAWICKI

We remark that such irrational numbers are precisely those that are badly approximable, that is, there exists c > 0 such that for all positive integers p, q we have
|α − p/q| > c/q 2 . A concrete example is any algebraic number of degree 2. The
term “badly approximable” comes from the fact that such a quadratic approximation is considered a bad approximation – by the Hurwitz theorem, every irrational
α admits infinitely many rationals p/q with |α − p/q| < 1/q 2 .
Indeed, recall that we have the following formulae (see, for instance, [31, Chapter 6]):
pi = ai pi−1 + pi−2
(12)

qi = ai qi−1 + qi−2
pi
(−1)i
pi+1
−
=
qi+1
qi
qi qi+1

Hence, both sequences (pi ) and (qi ) are increasing (for α > 0). We also have
pi /qi < α for even i and pi /qi > α for odd i, so we can bound:
(13)

α−

pi
pi
1
pi+1
<
−
=
.
qi
qi+1
qi
qi qi+1

Proof of Theorem A.1. Let α = [a0 , a1 , . . .] ∈ (0, 1) be an irrational number. Since
q0 = 1, for every l0 ∈ [1, ∞) there exists i such that qi ≤ l0 and qi+1 > l0 . Then,
by (13) we have |qi α − pi | ≤ 1/qi+1 < 1/l0 .
By (12) we have
qi = ai qi−1 + qi−2 ≤ (ai + 1)qi−1 ,
so, when A = supi ai is√finite, we see that qi > l0 /(A + 1). Consequently, for every
t ≥ 1 there exists l0 = t and i such that t/(A + 1) ≤ l0 qi ≤ t and |qi α − pi | ≤ 1/l0 .
If w = t/(l0 qi ), then, for l = wl0 , we have t = lqi and
|qi α − pi | ≤ w/wl0 = w/l ≤ (A + 1)/l.

That is, for every t ≥ 1 there exist l, p, q satisfying q ≤ l, |qα − p| ≤ (A + 1)/l
and √
lq = t. Theorem A.1 follows from Lemma A.5 (for m = 1), because the torus
Lipschitz equivalent to the torus (T1 , ld) × (T1 , qd), when
(T2 , √td) is (uniformly)
√
l = w t and q = t/w for w ∈ [1, A + 1).
Note that when α is rational, then the warped cone is quasi-isometric to a collection of circles (by Example 4.11 item (1) or [15]), which is clearly not quasi-isometric
to a collection of tori (for instance, they have different asymptotic dimension).
When α is irrational, but not restricted, that is, A = ∞, then qi+1 /qi is unbounded as a sequence in i. Consequently, when we put m = 1, l = qi+1 and
q = qi in Lemma A.5, then the respective levels are quasi-isometric to the more
and more “unbalanced” tori (T2 , qi+1 d×qi d). They cannot be (C, B)-quasi-isometric
to ordinary tori (T2 , ti d).
Indeed, quasi-isometry approximately preserves diameters, that is, any torus
(T2 , ti d) quasi-isometric to (T2 , qi+1 d × qi d) must satisfy (say that we are using the
ℓ∞ -metrics):
C −1 ti − 2B ≤ qi+1 ≤ Cti + 2B
for some C ≥ 1, B ≥ 0 not depending on i. For N > 0, consider the following invariant vN of metric spaces – vN (X) is the maximal cardinality of an N -separated subset of X. Since the Riemannian volume of (T2 , ti d) is t2i and for (T2 , qi+1 d × qi d) it
equals qi+1 qi , we also have vN (T2 , ti d) ≍ t2i /N 2 and vN (T2 , qi+1 d×qi d) ≍ qi+1 qi /N 2
for N ≤ ti , qi , where ≍ denotes equality up to some universal multiplicative constants. If there exists a (C, B)-quasi-isometric embedding
(T2 , ti d) → (T2 , qi+1 d × qi d),

WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

27

then vN (T2 , ti d) ≤ vN/C−B (T2 , qi+1 d × qi d), which contradicts the fact that
lim inf (qi+1 qi /t2i ) = 0.
i



Kim proved that if two continued fractions α = [ai ], β = [bi ] have the same
“tail”, that is, there exists k, l ∈ N such that ak+i = bl+i for all i ∈ N, then the
respective warped cones are quasi-isometric [15]. This tail equivalence relation gives
a classification of rotation C∗ -algebras C(T1 ) ⋊α Z up to Morita equivalence. Kim
asked whether the tail relation also provides the quasi-isometric classification of
warped cones (over actions by rotations). In particular, the affirmative answer to
her question would imply that each set of angles α giving a quasi-isometric warped
cone is countable; note that this is indeed the case for rational angles α, since they
form such a set [15]. Theorem A.1 identifies another set of this kind, which can
be thought of as another step towards a classification (second step out of infinitely
many...). Our result shows that countability does not hold in general, and hence
gives a negative answer to Kim’s question.
Corollary A.7. There exist uncountably many angles α ∈ (0, 1) yielding the same
quasi-isometry type of a warped cone over the rotation by α. In particular, the
equivalence relation on (0, 1) given by quasi-isometry of warped cones (over the respective rotations) is strictly coarser than the tail equivalence relation for continued
fractions.
The following remark shows that all warped cones over irrational rotations have
something in common, even though, by a result of Kim, there are infinitely many
quasi-isometry types among them.
Remark A.8. Even when the number α is not restricted, we still have a quasiisometry on a subsequence (qi2 T1 , dZ )i∈N ≃ (qi T2 , d)i∈N . Indeed, by (13) it suffices
to apply Theorem A.5 to numbers m = 1, l = qi and q = qi .
Proof of Corollary A.2. Fix t > 0. For any metric space (X, d) with an action of a
group generated by two elements r, ε we have the equality of warped metrics
dhε,ri = (dhεi )hri ,
where the respective actions of hεi and hri are given by restrictions of the action of
hε, ri, and both metrics are considered on tX.
Consider the following generators of D∞ : r and ε = r′ r. It follows that dD∞ =
(dZ )hri , where dZ is the metric on tT1 as in Theorem A.1 where Z = hεi. Let us
assume for simplicity that r acts by complex conjugation if we consider T1 as a
subset of C.
Since we have a bi-Lipschitz equivalence of dZ and d′Z (as in the proof of Lemma
A.5), we get a bi-Lipschitz equivalence (dZ )hri ≃ (d′Z )hri .
Consider the action of hri on T2 by complex conjugation on both coordinates
(via the inclusion T2 ⊆ C2 ). Clearly, ι : tT1 → T2 from the proof of Lemma A.5
is equivariant under the action of hri. Since it is also a√bi-Lipschitz embedding
with respect to metric d′Z on tT1 and τ d on T2 for τ = t, we get a bi-Lipschitz
equivalence (tT1 , (d′Z )hri ) ≃ (τ im(ι), dhri ). However, since B-neighbourhood of
im(ι) is the whole τ T2 for B = (A + 1)/2 (we proved that for metric D from the
proof of Lemma A.5 and B = 1/2; metric D is (A + 1)-Lipschitz equivalent to
metric τ d, so our claim holds for B = (A + 1)/2 and metric τ d, and consequently
for the warped metric dhri on τ T2 ), the inclusion (τ im(ι), dhri ) ⊆ (τ T2 , dhri ) is a
quasi-isometry. (Note that, by Lemma 2.5, the warped metric on im(ι) and the
restriction to im(ι) of the warped metric on T2 are equal, because the action is
isometric.)

28

DAMIAN SAWICKI

The last point is to observe that (τ T2 , dhri ) is quasi-isometric to (τ S2 , d) (where
d is, say, the standard geodesic metric on S2 ). This follows from Example 4.11
item (1), and the fact that the quotient of T2 by the above action is bi-Lipschitz
equivalent to S2 .
For the “only if” part, observe that if ε acts by a rational rotation, then the action
factors through a finite group and the respective quotient (see Example 4.11 item
(1)) is an interval; no unbounded family of intervals is however quasi-isometric to a
family of spheres. When ε acts by an irrational rotation, one can reason similarly as
in the proof of Theorem A.1, because the quotient of (T2 , ad×bd) by the action of hri
has diameter comparable with the diameter of (T2 , ad × bd), that is, approximately
max(a, b), and the same applies to the invariant vN from the proof of Theorem
A.1 (which is approximately ab/N 2 ). However, the vN invariant of the sphere of
diameter c is roughly c2 /N 2 .

Note that the actions in Corollary A.2 are the only interesting isometric actions
D∞ yT1 . If one of the generators acts by a rotation, then the action factors through
the finite group Z/2Z × Z/2Z.
Finally, let us prove Proposition A.3.
Proof of Proposition A.3. Fix m ∈ {2, 3, 4, . . .}. Choose b1 ≤ . . . ≤ bm odd primes.
Let
∞

X


Ni,n
for Di,n = exp log(bi ) m2n logbi (2)
αi =
Di,n
n=1

where Ni,n ∈ {1, . . . , bi − 1} (since bi is greater than 2, there are uncountably many
such real numbers αi ). In other words, denominators Di,n are the largest powers
2n
of bi bounded by 2m .
Pk Ni,n
provides an approximation of αi
Fix k ∈ N+ . The partial sum βi,k = n=1 Di,n
2k+2

2n

with error bounded by 2b2i /2m
. Indeed, if Di,n were defined as 2m , then using
2n
2k+2
+(n−k−1)
the bound 2m ≥ 2m
for every n ≥ k + 1 and the formula for the sum
2k+2
of a geometric series we would bound the error by 2bi /2m
, and by definition
2n
we have Di,n ≥ 2m /bi . We express every βi,k as a fraction with denominator
Q
2k
qk = i Di,k and numerator pi,k . Since Di,k ≤ 2m , we have
 2k m
2k+2
2k+2
|qk αi − pi,k | ≤ qk · 2b2i /2m
≤ 2m
· 2b2i /2m
=

2k

(2

2b2i
2b2m
≤
,
)m−1
2m2k

m2k+1

that is, for n = 2m , q = qk , pi = pi,k and K = 2b2m , the first bullet point in
Lemma A.5 is satisfied.
Let us verify the second one. The role of li will be played by Di,k . Note that,
since Ni,k is invertible modulo bi , the number βQ
i,k multiplied
Q by Di,k generates
Z/Di,k Z. The same is true if we multiply βi,k by j Dj,k as j6=i Dj,k is invertible
Q
modulo Di,k . Note that βi,k · j Dj,k = pi,k . Finally, by the Chinese RemainL
Q
Q
der theorem, Z/( j Dj,k )Z admits a decomposition Z/( j Dj,k )Z ∼
= i Z/Di,k Z,
Q
where Z/Di,k Z is generated by the image of pi,k ∈ Z/ j Dj,k Z.
By Lemma A.5, for the action Zm yα T1 , the space (tY, dZm ) with t = nq =
Q
2k
2k Q
1 m2k
d) × i (T1 , Di,k d). But since 2m /bi ≤
2m
i Di,k is quasi-isometric to (T , 2
2k
2k
Di,k ≤ 2m , the latter is bi-Lipschitz equivalent to (Tm+1 , 2m d).


WARPED CONES, (NON-)RIGIDITY, AND PIECEWISE PROPERTIES

29

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Dawid Kielak
Fakultät für Mathematik
Universität Bielefeld
Postfach 100131
D-33501 Bielefeld
Germany
Damian Sawicki
Institute of Mathematics
Polish Academy of Sciences
Śniadeckich 8
00-656 Warszawa
Poland

dkielak@math.uni-bielefeld.de

www.impan.pl/~dsawicki/

