Flats in spaces with convex geodesic bicombings

arXiv:1508.02745v2 [math.MG] 14 Mar 2016

Dominic Descombes & Urs Lang∗
March 2, 2016

Abstract
In spaces of nonpositive curvature the existence of isometrically
embedded flat (hyper)planes is often granted by apparently weaker
conditions on large scales. We show that some such results remain valid
for metric spaces with non-unique geodesic segments under suitable
convexity assumptions on the distance function along distinguished
geodesics. The discussion includes, among other things, the Flat Torus
Theorem and Gromov’s hyperbolicity criterion referring to embedded
planes. This generalizes results of Bowditch for Busemann spaces.

1

Introduction

The geometry of spaces of global nonpositive curvature is largely dominated
by the convexity of the distance function. Thus a considerable part of the
theory of CAT(0) spaces [2, 7] carries over to Busemann spaces [8, 27] (defined by the property that d◦(σ1 , σ2 ) is convex for any pair of constant speed
geodesics σ1 , σ2 parametrized on the same interval). However, this larger
class of spaces has the defect of not being preserved under limit processes.
For example, among normed real vector spaces, exactly those with strictly
convex norm satisfy the Busemann property, and a sequence of such norms
on Rn , say, may converge to a non-strictly convex norm. This motivates
the study of an even weaker notion of nonpositive curvature that dispenses
with the uniqueness of geodesics but retains the convexity condition for a
suitable selection of geodesics (compare Sect. 10 in [23]). In any normed
space, the affine segments t 7→ (1 − t)x + ty (t ∈ [0, 1]) provide such a choice.
In particular, the relaxed condition carries the potential for simultaneous
generalizations of results for nonpositively curved and Banach spaces. Another reason for this investigation is that l1 - and l∞ -type metrics have been
put in use in geometric group theory; see, for example, [3, 5, 9, 24]. The
recent paper [22] shows that symmetric spaces of noncompact type possess
natural, non-strictly convex Finsler metrics adapted to the geometry of Weyl
chambers and pertinent to the dynamics at infinity.
∗

Research supported by the Swiss National Science Foundation.

1

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D. Descombes & U. Lang

In a previous article we initiated a systematic study of spaces of weak
global nonpositive curvature as described above, with the main objective of
providing geometric models of this type for hyperbolic groups; see [11] and
in particular Theorem 1.3 therein. The purpose of the present paper is to
carry the analogy with CAT(0) and Busemann spaces further with regard
to existence results for flat subspaces. Here, for a metric space X = (X, d),
a map σ : X × X × [0, 1] → X will be simply called a bicombing if the
respective family of maps σxy := σ(x, y, ·) : [0, 1] → X satisfies the following
three properties:
(i) σxy is a geodesic from x to y, that is, σxy (0) = x, σxy (1) = y, and
d(σxy (t), σxy (t′ )) = |t − t′ | d(x, y) for t, t′ ∈ [0, 1];
(ii) σyx (t) = σxy (1 − t) for t ∈ [0, 1];
(iii) d(σxy (t), σx′ y′ (t)) ≤ (1 − t) d(x, x′ ) + t d(y, y ′ ) for t ∈ [0, 1].
(This corresponds to a conical and reversible geodesic bicombing in the
terminology of [11].) Notice that these conditions do not ensure that t 7→
d(σxy (t), σx′ y′ (t)) is a convex function on [0, 1]. This is guaranteed under the
following extra assumption on the traces:
(iv) im(σpq ) ⊂ im(σxy ) whenever p = σxy (r) and q = σxy (s) with r ≤ s.
Note that then σpq (t) = σxy ((1 − t)r + ts) for t ∈ [0, 1] by (i). A bicombing σ
satisfying (iv) will be called consistent. Busemann spaces and convex subsets
of normed spaces possess consistent bicombings, whereas some additional
examples of (general) bicombings are obtained via 1-Lipschitz retractions
onto subspaces. We refer to [11] for more information.
Our first main result is the following generalization of the hyperbolicity
criterion for cocompact CAT(0) spaces stated on p. 119 in [20]. A detailed
proof of Gromov’s result, inspired by [13], was given in [6]. For the case of
Busemann spaces, both Theorem 1.1 and Theorem 1.2 below were shown by
Bowditch [4].
Theorem 1.1 (Flat plane). Let X be a proper metric space with a consistent
bicombing σ and with cocompact isometry group. Then X is hyperbolic if
and only if X does not contain an isometrically embedded normed plane.
Another well-known result from the theory of spaces of nonpositive curvature is the Flat Torus Theorem, originally proved for smooth manifolds
in [19, 25] (see also [28, 14] for some earlier contributions in this direction). A
detailed account of this result and its applications in the context of CAT(0)
spaces is given in Chap. II.7 of [7]. We have:
Theorem 1.2 (Flat torus). Let X be a proper metric space with a consistent
bicombing σ. Let Γ be a group acting properly and cocompactly by isometries on X, and suppose that σ is Γ-equivariant. If Γ has a free abelian

Flats in spaces with convex geodesic bicombings

3

subgroup group A of rank n ≥ 1, then X contains an isometrically embedded
n-dimensional normed space on which A acts by translations.
Here, σ being Γ-equivariant means that γ ◦ σxy = σγ(x)γ(y) for all γ ∈ Γ
and (x, y) ∈ X × X. For example, beyond uniquely geodesic spaces, every
injective metric space (or absolute 1-Lipschitz retract) X admits a bicombing σ that is equivariant with respect to the full isometry group Isom(X)
of X; see Proposition 3.8 in [24]. Furthermore, it is shown in [11] that every proper metric space X with a bicombing and with finite combinatorial
dimension in the sense of [12] also admits a unique consistent bicombing,
which is Isom(X)-equivariant, too.
We briefly introduce some terminology that will be used throughout
the paper. Let X be a metric space. A map ξ : I → X of some interval
I ⊂ R is a geodesic if there is a constant c ≥ 0, the speed of ξ, such that
d(ξ(t), ξ(t′ )) = c|t − t′ | for all t, t′ ∈ I. A line or a ray in X is a unit
speed geodesic defined on R or R+ := [0, ∞), respectively. Two lines ξ, ξ ′
are parallel if sups∈R d(ξ(s), ξ ′ (s)) < ∞, and two rays η, η ′ are asymptotic if
sups∈R+ d(η(s), η ′ (s)) < ∞. A family of geodesics ξa : Ia → X indexed by a
set A will be called coherent if t 7→ d(ξa (α(t)), ξb (β(t))) is a convex function
on [0, 1] whenever a, b ∈ A and α : [0, 1] → Ia and β : [0, 1] → Ib are affine
maps1 . Notice that if σ is a consistent bicombing on X and A ⊂ X ×X is any
set, then {σxy : (x, y) ∈ A} is a coherent family. Given a bicombing σ on X,
we shall often write [x, y](t) for σxy (t) and [x, y] for im(σxy ) without further
comment. A set C ⊂ X will be called σ-convex if [x, y] ⊂ C whenever
x, y ∈ C. The (closed) σ-convex hull of a subset S ⊂ X is the smallest
(closed) σ-convex set containing S. A line ξ : R → X will be called a σ-line
if its trace is σ-convex; equivalently, [ξ(r), ξ(s)](t) = ξ((1 − t)r + ts) for all
r, s ∈ R and t ∈ [0, 1]. For two σ-lines ξ, ξ ′ the function s 7→ d(ξ(s), ξ ′ (s)) is
convex, hence constant in case ξ, ξ ′ are parallel.
The paper is organized as follows. In Sect. 2 we discuss a generalized
Flat Strip Theorem. Unlike for Busemann spaces, the σ-convex hull of a
pair of parallel σ-lines may be “thick” and the two lines may span different,
though pairwise isometric, flat (normed) strips. We also give a criterion
for the existence of an embedded normed half-plane. This is then used in
Sect. 3 for the proof of Theorem 1.1. A variant of this result for injective
metric spaces is also shown. In Sect. 4 we establish basic properties of semisimple isometries of spaces with bicombings. We employ a barycenter map
for finite subsets which was introduced in the context of Busemann spaces
in [15]. Sect. 5 addresses the question whether a hyperbolic (axial) isometry
of a metric space with a consistent bicombing σ also possesses an axis that
is at the same time a σ-line. This is false in general but holds true for the
hyperbolic elements of a group Γ as in Theorem 1.2. As an auxiliary tool we
use a fixed point theorem for nonexpansive mappings proved originally for
1

α, β are neither assumed to be surjective nor orientation preserving.

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D. Descombes & U. Lang

Banach spaces in [18]. Finally, Sect. 6 contains the proof of Theorem 1.2,
and we conclude by an example in which the embedded flat cannot be chosen
to be σ-convex.
In a subsequent paper [10] by the first author it is shown that a proper cocompact metric space X with a (possibly non-consistent) bicombing contains
an isometric copy of the n-dimensional normed space V under the following
asymptotic condition, also studied in [29]: there exist subsets Sk ⊂ X and
a sequence 0 < Rk → ∞ such that the rescaled sets (Sk , Rk−1 d) converge in
the Gromov–Hausdorff topology to the unit ball of V . This generalizes a
result of Kleiner for Busemann spaces; see Proposition 10.22 and the more
comprehensive Theorem D in [23]. Likewise, it follows that a proper cocompact metric space X with a bicombing contains a flat (normed) n-plane
whenever there is a quasi-isometric embedding of Rn into X, a result which
was first shown for manifolds of nonpositive curvature in [1]. Using these
more recent results from [10] one can extend Theorem 1.1 and Theorem 1.2,
except possibly for the fact that the subgroup A acts on the embedded flat,
to the case of general bicombings. It is not clear, however, whether this
yields significant improvements. In fact, it is still an open problem whether
there exists a metric space, proper or not, that admits a bicombing but no
consistent bicombing. In any case, the arguments presented here are more
direct. If the (consistent) bicombing σ in Theorem 1.1 is equivariant with
respect to a cocompact group of isometries of X, and X is non-hyperbolic,
then the construction we describe produces an embedded normed plane that
is foliated by mutually parallel σ-lines in at least one direction.

2

Flat strips and half-planes

We start with the construction of an embedded flat strip in an arbitrary
metric space, using only a minimal coherent family of geodesics, as defined
in the introduction.
Proposition 2.1 (Flat strip). Let X be a metric space. Suppose that
{ξ, ξ ′ } ∪ {ηs : s ∈ R} is a coherent collection of geodesics in X, where ξ, ξ ′
are two parallel lines with disjoint images and ηs : [0, 1] → X is a geodesic
from ξ(s) to ξ ′ (s) for every s ∈ R. Then the map
f : R × [0, 1] → X,

f (s, t) = ηs (t),

is an isometric embedding with respect to the metric on R × [0, 1] induced by
some norm on R2 .
Proof. For r ∈ R, put ν(r) := d(ξ(0), ξ ′ (r)). We have d(ξ(R), ξ ′ (R + r)) =
ν(r) for every R ∈ R since the left hand side is a convex non-negative
bounded function of R, hence constant. We claim that for every pair of

Flats in spaces with convex geodesic bicombings

5

points p = (s, t) and p′ = (s + ∆s, t + ∆t) in R × [0, 1] we have
′

d(f (p), f (p )) =

(

|∆t| ν
|∆s|

∆s
∆t



if ∆t 6= 0,
if ∆t = 0.

(2.1)

There is no loss of generality in assuming ∆t ≥ 0. Suppose first that ∆t > 0,
′
and put r := ∆s
∆t . Let q := (s − tr, 0) and q := (s + (1 − t)r, 1) denote the
′
points where the line through p, p intersects R × {0} and R × {1}. Then
d(f (q), f (q ′ )) = d(ξ(s − tr), ξ ′ (s − tr + r)) = ν(r).

(2.2)

We put η := ηs and η ′ := ηs+∆s . By convexity, we get that
d(f (q), f (p)) = d(ξ(s − tr), η(t)) ≤ t d(ξ(s − r), η(1)) = t ν(r).

(2.3)

Likewise, we have
d(f (p′ ), f (q ′ )) ≤ (1 − t − ∆t) ν(r)

(2.4)

as well as d(η(0), η ′ (∆t)) ≤ ∆t ν(r) and d(η(1−∆t), η ′ (1)) ≤ ∆t ν(r). Hence,
by the convexity of λ 7→ d(η(λ), η ′ (λ + ∆t)) on [0, 1 − ∆t], also
d(f (p), f (p′ )) = d(η(t), η ′ (t + ∆t)) ≤ ∆t ν(r).

(2.5)

From (2.2)–(2.5) and the triangle inequality we see that all inequalities derived so far are in fact equalities. In view of (2.5), this shows in particular
the first part of (2.1). The second case follows by continuity from the first,
since |r| − ν(0) ≤ ν(r) ≤ ν(0) + |r| for all r ∈ R and hence
lim |∆t| ν

∆t→0

∆s
∆t



= |∆s|.

Now, to conclude the proof, note that ν(r) > 0 for all r ∈ R, as ξ and ξ ′
have disjoint images. It then follows readily from (2.1) that there is a norm
k · k on R2 such that d(f (p), f (p′ )) = kp′ − pk for all p, p′ ∈ R × [0, 1]. Note
that the triangle inequality for k · k is just inherited from X.
The following example shows that, in general, if we replace ξ by s 7→
ξ(s + a) for some a 6= 0, we may get a different strip in X.
Example 2.2. Define piecewise affine functions g, ḡ : R × [0, 1] → R such
that

 12 t
if s ≤ 0,

1
g(s, t) = 2 |s − t| if 0 ≤ s ≤ 1,

1
2 (1 − t) if s ≥ 1,

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D. Descombes & U. Lang

and ḡ(s, t) = 21 − g(s, 1 − t). Note that g = ḡ outside of (0, 1)2 , whereas
the graphs of g and ḡ over [0, 1]2 bound a simplex in R3 . Consider the sets
Y = R × [0, 1] × R and
X := {(s, t, u) ∈ Y : g(s, t) ≤ u ≤ ḡ(s, t)},
both equipped with the metric induced by the maximum norm on R3 . Let
σ̄ : (x, y, λ) 7→ (1 − λ)x + λy be the canonical bicombing on Y . The vertical retraction π : (s, t, u) 7→ (s, t, min{max{u, g(s, t)}, ḡ(s, t)}) from Y onto
X is 1-Lipschitz. It follows from results in [11] (see Lemma 2.1, Theorem 1.1, the observation at the end of Sect. 2, and Theorem 1.2) that
σ̃ := π ◦ σ̄|X×X×[0,1] is a bicombing on X and that X possesses a unique
consistent bicombing σ, which furthermore satisfies σxy = σ̃xy whenever
the consistency condition (iv) stated in the introduction holds for σ̃xy . In
particular, the geodesics ξ, ξ ′ : R → X defined by ξ(s) := (s, 0, g(s, 0)) and
ξ ′ (s) := (s, 1, g(s, 1)) are two (parallel) σ-lines. It is then not difficult to see
that the strip formed by the segments σξ(s)ξ ′ (s+1) corresponds to the graph
of g, whereas the segments σξ(s+1)ξ ′ (s) trace out the graph of ḡ.
We also see that in Proposition 2.1, for fixed t ∈ (0, 1), the lines s 7→
f (s, t) need not be σ-lines in general: clearly the lines s 7→ σξ(s)ξ ′ (s+1) ( 12 ) and
s 7→ σξ(s+1)ξ ′ (s) ( 21 ) in the above example cannot both be σ-lines. However,
it is not difficult to deduce the following result.
Theorem 2.3 (Flat strips). Let X be a metric space with a consistent
bicombing σ, and let ξ, ξ ′ : R → X be two parallel σ-lines with disjoint traces.
Then there exists a norm on R2 such that the following assertions hold for
the metric it induces on R × [0, 1]:
(1) For every a ∈ R, the map fa : R × [0, 1] → X satisfying fa (s + ta, t) =
σξ(s)ξ ′ (s+a) (t) for all (s, t) ∈ R × [0, 1] is an isometric embedding.
(2) If, in addition, X is proper, there also exists an isometric embedding
f : R × [0, 1] → X such that f (·, 0) = ξ, f (·, 1) = ξ ′ , and s 7→ f (s, t) is
a σ-line parallel to ξ and ξ ′ for every fixed t ∈ (0, 1).
For a corresponding (but easier) result in the case of Busemann spaces,
see Lemma 1.1 and the remark thereafter in [4] (compare also Proposition 5.3
in [16]). Part (1) is a direct consequence of Proposition 2.1, and (2) then
follows by a simple limit argument (notice that in (1), all fa satisfy fa (·, 0) =
ξ and fa (·, 1) = ξ ′ ). As this result will not be used in the sequel, we omit
the details.
We now proceed to an existence result for embedded flat half-planes,
which will be instrumental in the proof of Theorem 1.1. We need the following analogue of the Tits cone in the case of CAT(0) spaces. Let R be a
coherent collection of rays in X, and denote by R(∞) the set of equivalence

Flats in spaces with convex geodesic bicombings

7

classes of mutually asymptotic rays in R. For (a, ξ), (b, η) ∈ R+ × R, we
put
1
d∞ ((a, ξ), (b, η)) := lim d(ξ(aλ), η(bλ)).
λ→∞ λ
Note that the limit exists by convexity, and |a−b| ≤ d∞ ((a, ξ), (b, η)) ≤ a+b.
This defines a pseudometric d∞ on R+ × R, and the respective quotient
metric space is a metric cone over R(∞) (compare [2], p. 38). In particular,
for a > 0, d∞ ((a, ξ), (a, η)) = a d∞ ((1, ξ), (1, η)), and this is zero if and only
ξ and η are asymptotic. The following result should now be compared with
Proposition II.4.2 in [2] and Proposition II.9.8 and Corollary II.9.9 in [7].
Proposition 2.4 (Flat half-plane). Let X be a metric space. Suppose that
{ξ} ∪ {ηs : s ∈ R} is a coherent collection of geodesics in X, where ξ is a
line and every ηs is a ray with ηs (0) = ξ(s) asymptotic to η := η0 . Then,
for all a, b > 0, the function s 7→ d(ξ(s + a), ηs (b)) is non-decreasing on R
with limit
lim d(ξ(s + a), ηs (b)) = d∞ ((a, ξ), (b, η)).
(2.6)
s→∞

Furthermore, if for every a ∈ R the function s 7→ d(ξ(s+a), ηs (1)) is constant
on R and nonzero, then the map
f : R × R+ → X,

f (s, t) := ηs (t),

is an isometric embedding with respect to the metric on R × R+ induced by
some norm on R2 .
Proof. Let a, b > 0. First we show that for all 0 < r ≤ λ ≤ r + 1,
d(ξ(ar + a), ηar (b)) ≥

1
d(ξ(aλ), η(bλ)).
λ

(2.7)

Since ηar and the ray t 7→ η(br + t) are asymptotic, we have
d(ηar (b), η(br + b)) ≤ d(ηar (0), η(br)) = d(ξ(ar), η(br)).

(2.8)

It follows that
d(ξ(ar + a), ηar (b)) ≥ d(ξ(ar + a), η(br + b)) − d(ξ(ar), η(br))


r+1
r
≥
−
d(ξ(aλ), η(bλ)),
λ
λ
which is (2.7). Putting λ = 1 we get d(ξ(ra + a), ηra (b)) ≥ d(ξ(a), η0 (b)).
Likewise, for all s ∈ R and 0 < r ≤ 1,
d(ξ(s + ra + a), ηs+ra (b)) ≥ d(ξ(s + a), ηs (b)),
so s 7→ d(ξ(s + a), ηs (b)) is non-decreasing. Furthermore, for all s ∈ R and
λ ≥ 1, we have
d(ξ(s + a), ηs (b)) ≤

1
d(ξ(s + aλ), ηs (bλ)).
λ

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D. Descombes & U. Lang

Together with (2.7), this gives (2.6).
For the second part of the proposition we have that s 7→ d(ξ(s+a), ηs (1))
is constant for every a ∈ R and that these values ν(a) := d(ξ(a), η0 (1)) are
all positive. We first claim that
d(ξ(s + ta), ηs (t)) = t ν(a)

(2.9)

for all t ≥ 0. The left hand side is convex as a function of t, thus it suffices
to show this equality for 0 ≤ t ∈ Z. For t = 0, 1, (2.9) clearly holds.
Consequently, by convexity, d(ξ(s + ta), ηs (t)) ≥ t ν(a) for all t > 1. The
reverse inequality for 1 < t ∈ Z follows by the triangle inequality since
d(ηs+ka (t − k), ηs+ka−a (t − k + 1)) ≤ d(ηs+ka (0), ηs+ka−a (1)) = ν(a)
for k = t, t − 1, . . . , 1 (compare (2.8)). Next, we claim that for every pair of
points p = (s, t) and p′ = (s + ∆s, t + ∆t) in R × R+ we have
(

|∆t| ν − ∆s
if ∆t 6= 0,
′
∆t
d(f (p), f (p )) =
|∆s|
if ∆t = 0,
similarly as in the proof of Proposition 2.1. To show this, suppose without
loss of generality that ∆t ≥ 0. Let first ∆t > 0, and put a := ∆s
∆t and
q := (s − ta, 0). Then (2.9) yields
d(f (p), f (p′ )) = d(ηs (t), ηs+∆s (t + ∆t))
≤ d(ηs (0), ηs+∆s (∆t))

= ∆t ν(−a)

as well as d(f (q), f (p)) = t ν(−a) and d(f (q), f (p′ )) = (t + ∆t) ν(−a). This
gives d(f (p), f (p′ )) = ∆t ν(−a), as claimed. The rest of the proof follows as
in Proposition 2.1.

3

Flat Planes

We now turn to Theorem 1.1. Recall that a metric space X is δ-hyperbolic,
for some constant δ ≥ 0, if for every quadruple (w, x, y, z) ∈ X 4 ,
d(w, y) + d(x, z) ≤ max{d(w, x) + d(y, z), d(w, z) + d(x, y)} + 2δ.
If such a δ exists, X is said to be hyperbolic. As is well known, for a geodesic
metric space this is equivalent to saying that geodesic triangles are slim, in
an appropriate sense. It also suffices to consider triangles whose sides are
given by a fixed bicombing:
Lemma 3.1. Let X be a metric space with a map that selects for every pair
of points x, y ∈ X a geodesic segment [x, y] = [y, x] ⊂ X connecting them.
If for every triple (x, y, z) ∈ X 3 the segment [x, z] is contained in the closed
δ
2 -neighborhood of [x, y] ∪ [y, z], then X is δ-hyperbolic.

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Flats in spaces with convex geodesic bicombings

Proof. Let (w, x, y, z) ∈ X 4 . The union of the closed 2δ -neighborhoods of
[x, y] and [y, z] contains [x, z]. Likewise, the closed 2δ -neighborhoods of [z, w]
and [w, x] cover [x, z]. It follows that there is either a pair of points x′ ∈ [x, y]
and z ′ ∈ [z, w] with d(x′ , z ′ ) ≤ δ or a pair of points y ′ ∈ [y, z] and w′ ∈ [w, x]
with d(y ′ , w′ ) ≤ δ. In the first case,


d(w, y) + d(x, z) ≤ d(w, z ′ ) + δ + d(x′ , y) + d(x, x′ ) + δ + d(z ′ , z)
= d(w, z) + d(x, y) + 2δ.

Similarly, in the second case, d(w, y) + d(x, z) ≤ d(w, x) + d(y, z) + 2δ.
In particular, a non-hyperbolic X with a bicombing σ contains a sequence of fatter and fatter σ-triangles. The following argument then uses
a ruled surface construction together with the cocompact isometric action
to produce a collection of mutually asymptotic rays as in Proposition 2.4.
This differs from the strategy in [4] and is inspired by the proof for CAT(0)
spaces in [6, 7], although we make no use of angles.
Proof of Theorem 1.1. Let X be a proper, cocompact metric space with a
consistent bicombing σ. If X contains an isometrically embedded normed
plane, then clearly X cannot be hyperbolic.
Suppose now that X is not hyperbolic. We show that then X must
contain an embedded normed plane. We continue to write [x, y] in place of
im(σxy ). By Lemma 3.1 there are sequences of points yn1 , yn2 , yn3 ∈ X and
pn ∈ [yn1 , yn3 ] such that
B(pn , n) ∩ ([yn1 , yn2 ] ∪ [yn2 , yn3 ]) = ∅

(3.1)

for all integers n ≥ 1, where B(pn , n) denotes the closed ball at pn of radius n. Put rn (·) := d(pn , ·). For i = 1, 2, let xin be a point in [yni , yni+1 ]
with minimal distance to pn , and let ξni : [0, rn (xin )] → X be a unit speed
parametrization of the segment [pn , xin ] from pn to xin . Then, for every pair
(i, j) ∈ {(1, 1), (1, 2), (2, 2), (2, 3)}, we define the “ruled surface”
i
j
∆i,j
n : [0, rn (xn )] × [0, rn (yn )] → X
i,j
i
i
so that ∆i,j
n (·, 0) = ξn and, for each s ∈ [0, rn (xn )], ∆n (s, ·) is a constant
j
speed parametrization of the segment [ξni (s), yn ] from ξni (s) to ynj . Thus
i,j
′
d(∆i,j
n (s, t), ∆n (s, t ))

=

d(ξni (s), ynj )
rn (ynj )

|t − t′ | ≤ 2|t − t′ |,

because d(ξni (s), ynj ) ≤ rn (xin ) + rn (ynj ) ≤ 2 rn (ynj ) by the choice of xin . Note
also that by convexity,
i,j ′
i
i ′
′
d(∆i,j
n (s, t), ∆n (s , t)) ≤ d(ξn (s), ξn (s )) = |s − s |.

(3.2)

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D. Descombes & U. Lang

2
It follows that each ∆i,j
n is 2-Lipschitz, where here and below we equip R
′
i
with the l1 -metric. Furthermore, putting s := rn (xn ), we notice that for
0 ≤ r ≤ s ≤ s′ and 0 ≤ t ≤ rn (ynj ),
i,j ′
i,j
i,j ′
d(ξni (r), ∆i,j
n (s, t)) ≥ rn (∆n (s , t)) − r − d(∆n (s, t), ∆n (s , t))

≥ rn (xin ) − r − (s′ − s)

=s−r

(3.3)

by the triangle inequality, the choice of xin , and (3.2).
Now we choose a sequence of isometries γn of X so that γn (pn ) ∈ K
for all n and for some fixed compact set K. By (3.1), rn (xin ), rn (ynj ) > n.
Since X is proper, we can extract a sequence n(k) so that each of the four
sequences γn(k) ◦ ∆i,j
n(k) converges uniformly on compact sets, as k → ∞, to
a 2-Lipschitz map
f i,j : R+ × R+ → X

with boundary rays ξ i := f i,j (·, 0) and η j := f i,j (0, ·). Furthermore, for
every s ∈ R+ , ηsi,j := f i,j (s, ·) is a ray asymptotic to η j , so f i,j is in fact
1-Lipschitz. Clearly {ξ i } ∪ {ηsi,j : s ∈ R+ } is a coherent collection of rays.
From the construction we also have that d(η 1 (t), η 3 (t)) = 2t for all t ≥ 0, in
particular η 1 , η 3 are non-asymptotic. Hence, there is at least one pair (i, j)
such that ξ i , η j are non-asymptotic. We put f := f i,j , ξ := ξ i , η := η j , and
ηs := ηsi,j for some such pair. We claim that for all a ∈ R and b > 0, the
limit
L(a, b) := lim d(ξ(s + a), ηs (b))
s→∞

exists and is strictly positive. Clearly L(0, b) = b. If a > 0, then L(a, b) =
d∞ ((a, ξ), (b, η)) > 0 by the first part of Proposition 2.4 and since ξ, η are
non-asymptotic. If a < 0, the same result still shows that s 7→ d(ξ(s +
a), ηs (b)) is non-increasing on [|a|, ∞), so the limit exists, and L(a, b) ≥ |a|
as a consequence of (3.3).
Next, for every integer l ≥ 1, we define the 1-Lipschitz map
fl : [−l, ∞) × R+ → X,

fl (s, t) := f (l + s, t) = ηl+s (t).

Then we choose isometries γ̄l of X so that (γ̄l ◦ fl )(0, 0) ∈ K for all l and
for some fixed compact set K. As above, there exists a subsequence l(k)
such that the sequence γ̄l(k) ◦ fl(k) converges uniformly on compact sets to
a 1-Lipschitz map
f¯: R × R+ → X
with boundary line ξ̄ := f¯(·, 0) and mutually asymptotic rays η̄s := f¯(s, ·)
for s ∈ R. Again, {ξ̄} ∪ {η̄s : s ∈ R} is a coherent collection of geodesics.
For every a ∈ R and b > 0, we now have that
d(ξ̄(s + a), η̄s (b)) = L(a, b) > 0

Flats in spaces with convex geodesic bicombings

11

for all s ∈ R. Hence, by Proposition 2.4, f¯ is an isometric embedding with
respect to some norm on R2 . Using once more that X is cocompact, we then
conclude that X contains an isometrically embedded normed plane.
It is clear that if X is a CAT(0) or a Busemann space, then this property
is inherited by any isometrically embedded normed plane, thus the corresponding norm must be Euclidean or strictly convex, respectively. We briefly
discuss another variant of Theorem 1.1, which happens to have a very short
proof, without reference to bicombings. Recall that a metric space X is
injective (as an object in the metric category with 1-Lipschitz maps as morphisms), if for every metric space B and every 1-Lipschitz map f : A → X
defined on a set A ⊂ B there is 1-Lipschitz extension f : B → X. By a
remarkable result of Isbell [21], every metric space Y has an injective hull
E(Y ), thus every isometric embedding of Y into an injective metric space
X factors as Y ⊂ E(Y ) → X (see Sects. 2 and 3 in [24] for a survey).
Let now Q = {w, x, y, z} be any metric space of cardinality four, and
suppose that c := d(w, y) + d(x, z) is not less than the maximum of
a := d(w, x) + d(y, z) and b := d(w, z) + d(x, y). The injective hull (or
the tight span [12]) of Q is isometric to the (possibly degenerate) rectangle
[0, 21 (c − a)] × [0, 21 (c − b)] in (R2 , k · k1 ) with four segments of appropriate
lengths attached at the corners, where the terminal points of these segments
correspond to Q. (See Fig. A1 on p. 336 in [12]. It is also worth pointing out
that the 1-skeleton of the tight span of Q, viewed as polyhedral complex, is
the unique optimal network realizing the metric of Q; see p. 325 in the same
paper.) Now the δ-hyperbolicity of Q means precisely that the width (the
minimum of the two side lengths) of this l1 -rectangle is not bigger than δ.
This has the following easy consequence.
Theorem 3.2. A proper, cocompact injective metric space X is hyperbolic
if and only if X does not contain an isometric copy of (R2 , k · k1 ) or, equivalently, of (R2 , k · k∞ ).
Proof. Suppose that X is not hyperbolic. Then, by the above observation,
for arbitrarily large δ > 0 there exists a quadruple Q ⊂ X whose injective
hull contains an isometric copy of [0, δ] × [0, δ] ⊂ (R2 , k · k1 ). Since X
is injective, this l1 -square embeds isometrically into X by the respective
property of the injective hull. From a sequence of such squares with side
lengths tending to infinity we obtain an isometric embedding of the entire
l1 -plane, using the fact that X is proper and cocompact.

4

Semi-simple isometries

In preparation for the proof of Theorem 1.2 we now discuss semi-simple
isometries of a metric space X with a (not necessarily consistent) bicombing σ. The main purpose is to establish basic properties regarding sets of

12

D. Descombes & U. Lang

minimal displacement, analogous to those in the case of CAT(0) spaces.
Whereas in the latter case a key role is played by the projection onto convex
subspaces, we use a barycenter map for finite subsets of X instead, which
we first describe. The same tool will be employed again in Sect. 6.
In [15], Es-Sahib and Heinich introduced an elegant barycenter construction for Busemann spaces, which was reviewed and partly improved in a
recent paper by Navas [26]. The construction and proofs translate almost
verbatim to spaces with bicombings. For finite subsets, the result is as
follows.
Theorem 4.1. Let X be a complete metric space with a bicombing σ. For
every integer n ≥ 1 there exists a map barn : X n → X such that
(1) barn (x1 , . . . , xn ) lies in the closed σ-convex hull of {x1 , . . . , xn };
P
(2) d(barn (x1 , . . . , xn ), barn (y1 , . . . , yn )) ≤ minπ∈Sn n1 ni=1 d(xi , yπ(i) );

(3) γ barn (x1 , . . . , xn ) = barn (γx1 , . . . , γxn ) whenever γ is an isometry of
X and σ is γ-equivariant.
We shall sometimes suppress the subscript n. The construction is such
that bar1 (x) := x, bar2 (x, y) := σxy ( 21 ) = σyx ( 12 ), and, for n ≥ 3,
barn (x1 , . . . , xn ) = barn (barn−1 (x1 ), . . . , barn−1 (xn )),
where xi := (x1 , . . . , xi−1 , xi+1 , . . . , xn ). The proof of Theorem 4.1 is then
not very difficult. The more profound observation of [15, 26] is that the above
construction can be modified so as to yield a barycenter map on the space
of probability measures with finite first moment that is 1-Lipschitz with
respect to the 1-Wasserstein metric. We will not use this more elaborate
construction in the present paper.
Now we turn to the discussion of isometries. We begin by recalling
some standard terminology and basic facts. First, let X be an arbitrary
metric space. For any map γ : X → X we denote by dγ (x) := d(x, γ(x)) the
displacement at a point x ∈ X, and we put
|γ| := inf dγ (x)
x∈X

and

Min(γ) := {x ∈ X : dγ (x) = |γ|}.

An isometry γ of X is called parabolic if Min(γ) is empty and semi-simple
otherwise. In the latter case, γ is elliptic if |γ| = 0 (that is, γ has a fixed
point) and hyperbolic if |γ| > 0.
For an isometry γ, a line ξ : R → X will be called an axis of γ if there
exists a t > 0 such that
γ(ξ(s)) = ξ(s + t) for all s ∈ R.

(4.1)

13

Flats in spaces with convex geodesic bicombings
Then, for x := ξ(0) and any y ∈ X, the triangle inequality gives
d(x, γ n (x)) ≤ d(x, y) + n dγ (y) + d(γ n (y), γ n (x))
= n dγ (y) + 2 d(x, y)

(4.2)

for all n ≥ 1, where d(x, γ n (x)) = nt, thus t ≤ dγ (y) and so dγ (x) = t = |γ|.
Hence every isometry γ with an axis is hyperbolic, and all axes of γ are
contained in Min(γ).
For the converse, let γ be a hyperbolic isometry of X with |γ| =: t, let
x ∈ Min(γ), and suppose there is a geodesic τ : [0, t] → X from x to γ(x).
Then the curve ξ : R → X satisfying
ξ(nt + s) = γ n (τ (s))

for all n ∈ Z and s ∈ [0, t]

(4.3)

is a local geodesic (in fact it preserves all distances less than or equal to t),
because ξ is parametrized by arc length and d(ξ(nt + s), ξ(nt + t + s)) =
dγ (τ (s)) ≥ t. This curve ξ also satisfies (4.1), hence it is an axis of γ if
it happens to be a line. This is the case, for example, if X is a Busemann
space, as then every local geodesic in X is a geodesic. (If η : [a, b] → X is the
geodesic from ξ(a) to ξ(b), then the nonnegative function s 7→ d(ξ(s), η(s))
is locally convex, hence convex on [a, b], hence identically zero as it vanishes
at the endpoints.) Thus every hyperbolic isometry of a Busemann space is
axial (compare Chap. 11 in [27]).
The following result shows in particular that this last fact remains true
in the more general context of this paper. Recall that a bicombing σ of
X is γ-equivariant, for an isometry γ of X, if γ ◦ σxy = σγ(x)γ(y) for all
(x, y) ∈ X 2 .
Proposition 4.2. Let γ : X → X be an isometry of a complete metric space
X with a γ-equivariant bicombing σ. Then:
(1) For all x, y ∈ X and n ≥ 1, |γ| ≤
(2) For all x ∈ X, limn→∞

1
n

1
n

d(x, γ n x) ≤ dγ (y) +

2
n

d(x, y).

d(x, γ n x) = |γ|.

(3) If γ is hyperbolic, then for every x ∈ Min(γ) there exists an axis of γ
through x.
(4) If C ⊂ X is non-empty, σ-convex, and γ-invariant, then |γ| = γ|C .
Proof. The second inequality in (1) is just (4.2). For the first, we employ
the barycenter construction stated above. Given x ∈ X and n ≥ 1, put
x := (x, γx, . . . , γ n−1 x) and γx := (γx, . . . , γ n x). Then
|γ| ≤ d(barn x, γ barn x) = d(barn x, barn γx) ≤

1
d(x, γ n x),
n

14

D. Descombes & U. Lang

where the last two steps use parts (3) and (2) of Theorem 4.1, respectively.
The limit formula (2) is an immediate consequence of (1). As for (3), let γ
be a hyperbolic isometry with |γ| =: t, and let x ∈ Min(γ). Then, for y = x,
(1) shows that d(x, γ n x) = nt for all n ≥ 1, thus any curve ξ as in (4.3) is
a line. Finally, given any set C as in (4), its closure C is still σ-convex and
γ-invariant, and furthermore complete. Now fix any x ∈ C and apply (2)
for both γ and γ|C to conclude that |γ| = γ|C . Clearly γ|C = γ|C .
The following standard result will be used several times in the sequel.
Lemma 4.3. Let X be a proper metric space, let Γ be a group acting properly and cocompactly by isometries on X, and let α1 , . . . , αn ∈ Γ. Given a
sequence of points in X along which the displacement functions dα1 , . . . , dαn
are bounded, there exist a subsequence xk and isometries γk ∈ Γ such that
γk xk converges to a point z ∈ X and, for every element α of the subgroup
hα1 , . . . , αn i, γk αγk−1 ∈ Γ is independent of k and limk→∞ dα (xk ) = dα (y)
for all points y in the sequence γk−1 z.
In particular, for any α ∈ Γ, starting from a minimizing sequence for dα
one gets that |α| = limk→∞ dα (xk ) = dα (y) for some point y. This shows
that Γ acts by semi-simple isometries (compare Proposition II.6.10 in [7]).
Proof. Since the action is cocompact we may assume, by passing to a subsequence xk , that there exist γk ∈ Γ such that γk xk converges to a point
z ∈ X. By assumption the sequence d(γk xk , γk α1 γk−1 (γk xk )) = dα1 (xk ) is
bounded, so d(z, γk α1 γk−1 z) is bounded as well. Hence, because the action
of Γ is proper, we may pass to a further subsequence in order to arrange
that γk α1 γk−1 is equal to the same element α1 ∈ Γ for all k. Repeating the
argument for α2 , . . . , αn , we arrive at a map αi 7→ αi which extends to a
homomorphism α → α from hα1 , . . . , αn i into Γ such that γk αγk−1 = α for
all k. Now it follows that
lim dα (xk ) = lim dα (γk xk ) = dα (z) = dα (y)

k→∞

k→∞

whenever y = γk−1 z for some k.
From the above results we obtain a crucial fact for the proof of Theorem 1.2.
Proposition 4.4. Let X be a proper metric space with a bicombing σ. Let Γ
be a group acting properly and cocompactly by isometries on X, and suppose
that σ is Γ-equivariant. Then for every finitely generated abelian subgroup
A of Γ the set
\
Min(A) :=
Min(α)
α∈A

is non-empty (and σ-convex, α-invariant for every α ∈ A, and closed).

Flats in spaces with convex geodesic bicombings

15

Proof. For an individual α ∈ A the set Min(α) is non-empty, as noted after
Lemma
4.3. Now suppose that B ⊂ A is a finite set such that Min(B) :=
T
β∈B Min(β) 6= ∅, and let α ∈ A \ B. Note that Min(B) is σ-convex and
furthermore α-invariant, as α commutes with every element of B. Using
Proposition 4.2(4) we find a sequence xk such that dα (xk ) → α|Min(B) = |α|
and dβ (xk ) = |β| for all β ∈ B. Applying Lemma 4.3 for the set B ∪ {α} ⊂ Γ
we get a point y ∈ Min(B ∪ {α}). This shows that Min(B) 6= ∅ for every
finite set B ⊂ A. Exhausting A by an increasing sequence of finite subsets
we obtain a sequence xk in X such that for every α ∈ A, the sequence
dα (xk ) is eventually constant with value |α|. Applying Lemma 4.3 again,
for generators α1 , . . . , αn of A, we conclude that Min(A) is non-empty.

5

σ-Axes

In Proposition 4.2 we showed that every hyperbolic isometry γ of a complete
metric space X with a γ-equivariant bicombing σ is axial. It is natural
to ask whether γ also admits an axis that is at the same time a σ-line.
Such an axis will be called a σ-axis. It turns out that the answer to this
question is negative in general, see Example 5.4. However, we shall prove in
Proposition 5.5 that any group Γ satisfying the assumptions of Theorem 1.2
acts by “σ-semi-simple” isometries, that is, every element has either a fixed
point or a σ-axis.
We start with an auxiliary result which will be useful in the proof of
Proposition 5.3. In [18], Goebel and Koter proved a fixed point theorem
for “rotative” nonexpansive mappings in closed convex subsets of Banach
spaces. The argument can easily be adapted to the present context.
Theorem 5.1. Let Y be a complete metric space with a bicombing σ. Then
every 1-Lipschitz map ϕ : Y → Y for which there exist an n ≥ 2 and 0 ≤
a < n such that
d(y, ϕn (y)) ≤ a d(y, ϕ(y))

for all y ∈ Y

has a fixed point. Furthermore, the fixed point set of ϕ is a 1-Lipschitz
retract of Y .
Proof. Let λ ∈ (0, 1) (an appropriate value depending only on a, n will be
determined at the end of the proof). For every x ∈ Y , the map sending y ∈ Y
to [x, ϕ(y)](λ) is λ-Lipschitz and thus has a unique fixed point fλ (x) ∈ Y
by Banach’s contraction mapping theorem. This yields a map fλ : Y → Y
with the property that fλ (x) = [x, ϕ(fλ (x))](λ) for all x ∈ Y . By convexity,
d(fλ (x), fλ (y)) ≤ (1 − λ) d(x, y) + λ d(ϕ(fλ (x)), ϕ(fλ (y)))
≤ (1 − λ) d(x, y) + λ d(fλ (x), fλ (y)),

16

D. Descombes & U. Lang

so fλ is 1-Lipschitz. Furthermore, fλ has the same fixed points as ϕ, because
[x, ϕ(x)](λ) = x if and only if ϕ(x) = x. By the assumption on ϕ,
d(y, ϕ(fλ (y))) ≤ d(y, ϕn (y)) + d(ϕn (y), ϕ(fλ (y)))

≤ a d(y, ϕ(y)) + d(ϕn−1 (y), fλ (y)).

(5.1)

To estimate the last term, note that, again by convexity,
d(ϕm (y), fλ (y)) ≤ (1 − λ) d(ϕm (y), y) + λ d(ϕm (y), ϕ(fλ (y)))
≤ (1 − λ)m d(ϕ(y), y) + λ d(ϕm−1 (y), fλ (y)),

for m ∈ {1, . . . , n − 1}. It follows that
d(ϕn−1 (y), fλ (y)) ≤ (1 − λ)cλ (n) d(ϕ(y), y) + λn−1 d(y, fλ (y)),

(5.2)

where cλ (n) := (n − 1) + (n − 2)λ + · · · + λn−2 . Combining the fact that
d(y, fλ (y)) = λ d(y, ϕ(fλ (y))) with (5.1) and (5.2) we get
d(y, fλ (y)) ≤

a + (1 − λ)cλ (n)
λ d(y, ϕ(y)).
1 − λn

Now let x ∈ Y , and put y := fλ (x). Then y = [x, ϕ(y)](λ) and hence
λ d(y, ϕ(y)) = (1 − λ) d(x, y), thus we obtain
d(fλ (x), fλ2 (x)) ≤

a + (1 − λ)cλ (n)
d(x, fλ (x)).
1 + λ + · · · + λn−1

Since the factor on the right converges to na < 1 for λ → 1, there is a
λ ∈ (0, 1) making it strictly less than 1. Then, for every x ∈ Y , the sequence
k 7→ fλk (x) is Cauchy. Since fλ is 1-Lipschitz, it follows that the limit point
̺(x) of this sequence is a fixed point of fλ , hence a fixed point of ϕ, and ̺
is a 1-Lipschitz retraction of Y onto the fixed point set of ϕ.
Now let γ be any isometry of a metric space X with a bicombing σ. We
associate with γ the map

ϕγ : X → X, ϕγ (x) = [γx, γ −1 x] 12 .

Note that d(ϕγ (x), ϕγ (y)) ≤ 21 d(γx, γy) + 21 d(γ −1 x, γ −1 y) = d(x, y), thus
ϕγ is 1-Lipschitz. Our interest in this map comes from the following simple
fact.
Lemma 5.2. Let γ be an isometry of a metric space X with a γ-equivariant
consistent bicombing σ, and let x ∈ X be such that γ(x) 6= x. Then there
exists a σ-axis of γ through x if and only if x is a fixed point of the associated
map ϕ = ϕγ .

17

Flats in spaces with convex geodesic bicombings

Proof. If ξ : X → R is a σ-axis of γ through x, then clearly ϕ(x) = x.
Conversely, suppose that x is a fixed point of ϕ. Put t := dγ (x). Let
τ : [0, t] → X be defined by τ (s) = [x, γx]( st ), and consider the corresponding
unit speed curve ξ : R → X satisfying (4.3). Since ϕ(x) = x and σ is
consistent, it follows that ξ is a “local σ-line”, in fact every subsegment of
length t is σ-convex. Then, as in the case of Busemann spaces, it follows
that ξ is a (global) σ-line and hence a σ-axis of x.
We now show that the translation length |ϕγ | = inf x∈X dϕγ (x) of ϕγ is
always zero, provided the bicombing σ is γ-equivariant.
Proposition 5.3. Let γ be an isometry of a metric space X with a γequivariant bicombing σ. Then for all x ∈ X and n ≥ 1,
√
d(x, ϕγ n (x)) ≤ n dγ (x),
and |ϕγ | = 0.
Proof. We assume without loss of generality that X is complete. We write
ϕ := ϕγ . Let x ∈ X. For all m ∈ Z and 0 ≤ n ∈ Z, put xn,m := ϕn (γ m x) and
dn,m := d(x, xn,m ). Note that ϕ and γ commute because σ is γ-equivariant.
In particular, for n ≥ 1, we have xn,m = [xn−1,m−1 , xn−1,m+1 ]( 12 ) and hence
1
dn,m ≤ (dn−1,m−1 + dn−1,m+1 ).
2
By induction on n this yields
−n

dn,m ≤ 2

n  
X
n
i=0

i

d0,m−n+2i .

Since d0,m ≤ |m| dγ (x), we obtain
−n

dn,0 ≤ 2

n  
X
n
i=0

i

−n

d0,2i−n ≤ 2 dγ (x) · 2

n  
X
n
i=0

i

i−

n
.
2

n

As 2−n i is the probability mass function of a binomial distribution with
parameters n and 21 (number of trials and probability of success), let Z be a
random variable distributed accordingly. Recall that the mean and variance
are E[Z] = n2 , Var[Z] = n4 , hence
i
hp
dn,0
(Z − E[Z])2
≤ E[|Z − E[Z]|] = E
2 dγ (x)
√
p
p
n
2
≤ E[(Z − E[Z]) ] = Var[Z] =
2
√
by Jensen’s inequality. Thus d(x, ϕn (x)) = dn,0 ≤ n dγ (x).

18

D. Descombes & U. Lang

For any c > |γ|, Y := {x ∈ X : dγ (x) ≤ c} is a non-empty, complete and
σ-convex set with γ(Y ) = Y and, consequently, ϕ(Y ) ⊂ Y . Now if |ϕ| was
positive, then for some sufficiently large n and for some a < n we would have
√
d(x, ϕn (x)) ≤ c n ≤ a|ϕ| ≤ a d(x, ϕ(x)) for all x ∈ Y , and Theorem 5.1
would provide a fixed point y = ϕ(y), in contradiction to |ϕ| > 0. This
shows that |ϕ| = 0.
The following example shows that, in general, the infimum |ϕγ | = 0 need
not be attained. The isometry γ we construct is axial, but has no σ-axis.
Example 5.4. Let X := l∞ (Z) be the Banach space of bounded functions
x : Z → R, with the supremum norm, and consider the affine bicombing
(x, y, λ) 7→ (1 − λ)x + λy (there is in fact no other bicombing on X, see
Theorem 1 in [17]). Let ̺ : X → X be the shift map satisfying ̺(x)(k) =
x(k − 1) for all x ∈ X and k ∈ Z, and let p ∈ X be defined by p(k) = 1 for
k ≥ 1 and p(k) = 0 otherwise. The isometry γ : X → X, γ(x) := ̺(x) + p,
satisfies kγ n (0)k∞ = n for all n ≥ 1, so |γ| = dγ (0) = 1 by Proposition 4.2.
Thus γ is hyperbolic and hence axial. The associated map ϕ = ϕγ : X → X
is given by
1
ϕ(x) = (̺(x) + ̺−1 (x) − z),
2
where z := ̺−1 (p) − p is the indicator function of 0. Now an x ∈ X with
ϕ(x) = x would have to fulfil x(0) = 21 (x(−1) + x(1) − 1) as well as x(k) =
1
2 (x(k −1)+x(k +1)) for all k 6= 0, and it is easy to see that no such bounded
function x : Z → R exists.
In contrast to this example, the following holds.
Proposition 5.5. Let X be a proper metric space with a consistent bicombing σ. Let Γ be a group acting properly and cocompactly by isometries on
X, and suppose that σ is Γ-equivariant. Then every isometry α ∈ Γ has
either a fixed point or a σ-axis.
Proof. Let α ∈ Γ. In view of Lemma 5.2 we just need to show that the associated map ϕ := ϕα has a fixed point. Let r > |α|. Applying Proposition 5.3
to the complete, σ-convex and α-invariant set Xr := {x ∈ X : dα (x) ≤ r},
we find a sequence of points in Xr along which the displacement function dϕ
tends to zero. By Lemma 4.3 there exist a subsequence xk and isometries
γk ∈ Γ such that γk xk converges to a point z ∈ X and γk αγk−1 =: α ∈ Γ
is constant. Put ϕ := ϕα . Since σ is γk -equivariant, we have that for all
y ∈ X,
γk ◦ [α−1 y, αy] = [α−1 (γk y), α(γk y)],
hence dϕ (y) = d(γk y, (γk ◦ ϕ)y) = dϕ (γk y). It follows that

dϕ (γk−1 z) = dϕ (z) = lim dϕ (γk xk ) = lim dϕ (xk ) = 0,
k→∞

thus every γk−1 z is a fixed point of ϕ.

k→∞

Flats in spaces with convex geodesic bicombings

6

19

Flat tori

We now prove Theorem 1.2. Thus, in the following, X denotes a proper
metric space with a consistent bicombing σ, equivariant with respect to a
group Γ that acts properly and cocompactly by isometries on X, and A
is a free abelian subgroup of Γ of rank n. We retain the multiplicative
notation for A ⊂ Γ, but we fix once and for all an isomorphism ι : (Zn , +) →
(A, ·). For generic points a, b ∈ Zn , the corresponding elements of A will
be denoted by α := ι(a), β := ι(b) without further comment. We write
b1 = (1, 0, . . . , 0, ), . . . , bn = (0, . . . , 0, 1) for the canonical generators of Zn
and put βi := ι(bi ). With this convention, we can state the assertion of
Theorem 1.2 as follows: there exist a norm k · k on Rn and an isometric
embedding f : (Rn , k · k) → X such that
αf (p) = f (p + a)

for all p ∈ Rn and a ∈ Zn .

(6.1)

This implies that d(f (p), αn f (p)) = knak for all n ≥ 1, therefore kak must
be equal to the translation length |α| by Proposition 4.2(2). We first show
that a norm with this latter property indeed exists. Notice that we already
know from Proposition 4.4 that Min(A) is non-empty.
Lemma 6.1. There is a unique norm k · k on Rn such that kak = |α| for
every a ∈ Zn . With respect to the metric on Zn induced by this norm, the
map a 7→ αx is an isometric embedding of Zn into X for every x ∈ Min(A).
Proof. Define kak := |α| for all a ∈ Zn . Then, for every x ∈ Min(A) and
a, b ∈ Zn ,
kb − ak = |α−1 β| = d(x, α−1 βx) = d(αx, βx),
and this is non-zero if α 6= β, for otherwise α−1 β would have a fixed point
and infinite order as A is free, in contradiction to the action being proper.
Furthermore, kmak = |m|kak for m ∈ Z because |αm | = |m||α| by Proposition 4.2. It follows that k · k extends uniquely to a norm on Qn and then
also to a norm on Rn .
In the following, Rn (and Zn , Qn ) are always equipped with the metric
induced by this norm k · k. The next result will constitute the last step of
the proof of Theorem 1.2.
Proposition 6.2. Assume that there exists a sequence of 1-Lipschitz maps
fk : Rn → X such that for all p ∈ Rn and i ∈ {1, . . . , n},
lim d(βi fk (p), fk (p + bi )) = 0.

k→∞

Then there is an isometric embedding f : Rn → X satisfying (6.1).

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D. Descombes & U. Lang

Proof. First note that the displacement of βi along the sequence fk (0) is
bounded by d(βi fk (0), fk (bi )) + d(fk (bi ), fk (0)), where the first term goes
to zero by assumption and the second is bounded by kbi k. By Lemma 4.3
we may assume, after passing to a subsequence, that there are isometries
γk ∈ Γ such that γk fk (0) converges to a point in X and γk αγk−1 =: α ∈ Γ
is constant for every α ∈ A. By the Arzelà–Ascoli theorem we may further
assume that the sequence of 1-Lipschitz maps γk ◦ fk converges uniformly
on compact sets to a 1-Lipschitz map h : Rn → X. Now, for all p ∈ Rn and
a ∈ Zn , we have that
d(αγk−1 h(p), γk−1 h(p + a)) = d(αh(p), h(p + a))
= lim d(αγk fk (p), γk fk (p + a))
k→∞

= lim d(αfk (p), fk (p + a)).
k→∞

By assumption this last term is zero if a ∈ {b1 , . . . , bn }. Thus, for any fixed
k, the map f := γk−1 ◦ h satisfies αf (p) = f (p + a) for all p ∈ Rn and for
all generators a = bi , hence for all a ∈ Zn . This property then forces the
1-Lipschitz map f to be isometric on Zn because
kak = |α| ≤ d(f (p), αf (p)) = d(f (p), f (p + a)) ≤ kak
for all p, a ∈ Zn . Furthermore, every line segment in Rn connecting two
points in Zn is embedded isometrically. Since the set of all pairs of points
which lie on a common such segment is dense in Rn × Rn , we conclude that
f is in fact an isometric embedding.
Now we proceed as follows. First we construct a 1-Lipschitz map
g : Rn → Min(A) that sends every ray R+ a with a ∈ Zn \ {0} isometrically to a (σ-)ray asymptotic to a σ-axis of α. Then we use a discrete
averaging process based on barycenters (Theorem 4.1) to find a sequence of
maps satisfying the assumptions of Proposition 6.2.
Proof of Theorem 1.2. We fix a point x ∈ Min(A) and define a map g : Rn →
X as follows. First, for a ∈ Zn \ {0} and λ ∈ [0, 1], put

g(λa) := lim [x, αk x] λk .
k→∞

The limit exists by Lemma 5.1 in [11] since the orbit hαix stays within finite
distance of some σ-axis of α by Proposition 5.5, and the definition is clearly
consistent for distinct representations of the same point. For a, b ∈ Zn \ {0}
and λ ∈ [0, 1] we have


d(g(λa), g(λb)) = lim d [x, αk x] λk , [x, β k x] λk
k→∞

λ
d(αk x, β k x)
k→∞ k
= λka − bk,
≤ lim

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Flats in spaces with convex geodesic bicombings

in particular g is 1-Lipschitz on Qn \ {0}. It follows that g extends uniquely
to a 1-Lipschitz map g : Rn → X.
Next, for all integers k ≥ 1, put Ik := [−k, k]n ∩Zn . We want to establish
the following estimate of sublinear growth:
e(k) := sup d(g(a), αx) = o(k)
a∈Ik

(k → ∞).

(6.2)

Given an ε > 0, there is a finite set B ⊂ Zn and a constant C such that for
every a ∈ Zn there exist a point b ∈ B and a positive integer m such that
ka − mbk ≤ εkak + C. For each b ∈ B we pick a point yb ∈ Min(A) on a
σ-axis of β. Then


d(g(mb), β m yb ) = lim d [x, β mk x] k1 , [yb , β mk yb ] k1 ≤ d(x, yb ),
k→∞

hence d(g(mb), β m x) ≤ 2 d(x, yb ). So let D := max{2 d(x, yb ) : b ∈ B}.
Now, for every a ∈ Zn , if b and m are as above, we have
d(g(a), αx) ≤ d(g(a), g(mb)) + d(g(mb), β m x) + d(β m x, αx)
≤ 2ka − mbk + D

≤ 2(εkak + C) + D.

This clearly yields (6.2).
To conclude the proof we now construct maps that meet the requirements
of Proposition 6.2. Define fk : Rn → X by
fk (p) := bar({α−1 g(p + a) : a ∈ Ik }).
Since g is 1-Lipschitz, it follows from Theorem 4.1(2) that fk is 1-Lipschitz
as well. For the generators bi we have
βi fk (p) = bar({α−1 g(p + bi + a) : a ∈ Ik − bi }),

fk (p + bi ) = bar({α−1 g(p + bi + a) : a ∈ Ik }),

the first equality being a consequence of Theorem 4.1(3) and a change of
variable. In order to estimate d(βi fk (p), fk (p + bi )) we need a pairing of
points in Ik with points in Ik − bi . We match a ∈ Ik ∩ (Ik − bi ) with itself
and a ∈ Ik \ (Ik − bi ) with ã := a − (2k + 1)bi ∈ (Ik − bi ) \ Ik . For a pair of
the latter type we have
d(α−1 g(p + bi + a), x) = d(g(p + bi + a), αx)
≤ d(g(p + bi + a), g(a)) + d(g(a), αx)
≤ kp + bi k + e(k)

as well as d(α̃−1 g(p + bi + ã), x) ≤ kp + bi k + e(k + 1), since Ik − bi ⊂ Ik+1 .
Thus
d(α−1 g(p + bi + a), α̃−1 g(p + bi + ã)) ≤ 2(kp + bi k + e(k + 1)).

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D. Descombes & U. Lang

Since there are (2k + 1)n−1 such pairs (a, ã) out of |Ik | = (2k + 1)n pairs in
total, we conclude that
d(βi fk (p), fk (p + bi )) ≤

2(kp + bi k + e(k + 1))
→ 0 (k → ∞)
2k + 1

by Theorem 4.1(2) and (6.2).
We conclude this section with an example illustrating Theorem 1.2.
Given X, σ, and an isometric embedding f : V → X of some n-dimensional
normed space V as in the theorem, f carries the canonical bicombing σ̄ on V
to a consistent bicombing on the image of f . However, the geodesics f ◦ σ̄pq
will in general not agree with σf (p)f (q) . In fact, the following example for
n = 2 shows that, despite of much flexibility in the construction of f , it may
happen that im(f ) is never σ-convex. This stands in contrast to the case
n = 1 treated in Proposition 5.5.
Example 6.3. Let w : R → R be the 1-periodic function satisfying w(t) = |t|
for t ∈ [− 12 , 12 ]. Define piecewise affine functions g, ḡ : R2 → R by g(s, t) :=
w(t) and ḡ(s, t) := max{w(s), w(t)}, and consider the set
X := {(s, t, u) ∈ R3 : g(s, t) ≤ u ≤ ḡ(s, t)},

endowed with the metric induced by the maximum norm on R3 . It follows
as in Example 2.2 that X admits a unique consistent bicombing σ and that
the lines ξ, ξ ′ : R → X defined by ξ(s) := (s, 0, 0) and ξ ′ (t) := ( 12 , t, 21 ) are
two σ-lines whose traces are contained in the graphs of g and ḡ, respectively. Clearly Z2 acts properly and cocompactly by isometries on X via
((z, z ′ ), x) 7→ x + (z, z ′ , 0), and the bicombing σ is Z2 -equivariant. Theorem 1.2 now implies that there exist a norm k · k on R2 and an isometric
embedding f : (R2 , k·k) → X such that f (p)+(z, z ′ , 0) = f (p+(z, z ′ )) for all
p ∈ R2 and (z, z ′ ) ∈ Z2 . To describe the image of f , let ̺ : X → (R2 , k · k∞ )
denote the 1-Lipschitz projection (s, t, u) 7→ (s, t). Since the third coordinates of any two points in X differ by at most 21 , it follows that ̺ ◦ f
preserves all distances greater than 12 , but this forces ̺ ◦ f to be an isometry
altogether. Hence ̺|im(f ) is an isometry as well, and this implies in turn
that im(f ) is the graph of a 1-Lipschitz function h : (R2 , k · k∞ ) → R such
that g ≤ h ≤ ḡ and h is Z2 -periodic. Now g = ḡ = 0 on Z2 and g = ḡ = 21
on R × ( 21 + Z). It follows in particular that the image of f contains the sets
ξ(Z) and ξ ′ ( 21 + Z) but cannot contain both the points ξ( 21 ) = ( 12 , 0, 0) and
ξ ′ (0) = ( 21 , 0, 12 ). Thus, no matter how f is chosen, the image of f will not
be σ-convex.

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D. Descombes (dominic.descombes@math.ethz.ch),
U. Lang (urs.lang@math.ethz.ch),
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland

