arXiv:1706.03743v2 [] 13 Jun 2017

Continuous cocycle superrigidity for the full shift over a finitely
generated torsion group.
David Bruce Cohen
June 14, 2017
Abstract
Chung and Jiang showed that, if a one ended group contains an infinite order element,
then every continuous cocycle over the full shift, taking values in a discrete group, must be
cohomologous to a homomorphism. We show that their conclusion holds for all one ended
groups, so that the hypothesis of admitting an infinite order element may be omitted.

1

Introduction.

Let A be a finite set equipped with the discrete topology, let G be a finitely generated group,
and let H be a discrete group. The full shift AG consists of all functions G→A equipped with
the product topology and the left G-action given by (gx)(h) = x(g −1 h). A result of Chung and
Jiang [1, Corollary 1] shows that when G is one ended (Definition 2.5) and contains an infinite order
element, AG has continuous cocycle rigidity—meaning that every continuous H-valued cocycle over
AG is cohomologous to some homomorphism from G to H, in the sense of the following definition.
Definition 1.1. (See [1, Definitions 2.1,2.2] and Figure 1.) Given a topological dynamical system
consisting of a group G acting by homeomorphisms on a space X, a continuous H-valued cocycle
over this system is a continuous function c : G × X → H such that c(gh, x) = c(g, hx)c(h, x). If
there exists a continuous function b : X→H and a homomorphism φ : G→H such that c(g, x) =
b(gx)φ(g)b(x)−1 , then c is said to be cohomologous to φ with transfer function b.
Continuous cocycles arise naturally in the study of continuous orbit equivalence [2] and topological couplings [3]. Our main theorem is that AG has continuous cocycle rigidity for every one ended
group G. In particular, our theorem holds even if G is periodic (has no infinite order element).
Theorem 1.2. If G is a one ended group, and H a discrete group, then every continuous cocycle
c : G × AG →H is cohomologous to a homomorphism.
See Theorem 4.5 for the proof. Our technique loosely follows that of [1], but we use geodesics
in the Cayley graph where [1] uses paths of the form n 7→ an (for a ∈ G having infinite order).

1.1

Outline

In §2 we recall some basic facts about the geometry of finitely generated groups. In particular, we
define geodesics and one-endedness, and prove the basic fact that every group contains a biinfinite
geodesic.
1

hx

c(g, x)

gx

c(h, x)

x

c(g, hx)

c(gh, x)

b(gx)

x

b(x)

φ(g)

ghx

Figure 1: At left, a van Kampen diagram depicting the cocycle identity. At right, a van Kampen
diagram depicting a cohomology from c to φ with transfer function b. Vertices are labeled so that
if an edge labeled h ∈ H connects a vertex labeled y ∈ X to a vertex labeled x ∈ X, then there is
some g ∈ G such that y = gx and h = c(g, x).
In §3 we prove Lemma 3.3, which states that, for a continuous cocycle c, and an appropriate L >
0, the value of c(g, x) is determined by c(h, x) together with the values of x in the L-neighborhood
of any path from h−1 to g−1 .
Proof strategy. Inspired by [1], in §4 we use the following strategy to prove our main theorem.
Suppose that G is a one ended group, H a discrete group, and c : G × AG →H a continuous cocycle.
Take 0 to denote some element of A.
• Define φ(g) = c(g, ~0) where ~0 ∈ AG is the configuration which is 0 everywhere.
• For x ∈ ∆(~0), take b(x) to be φ(gx )c(gx , x)−1 , where ∆(~0) ⊆ AG consists of configurations
with only finitely many non-0 entries, and gx ∈ G is chosen so that its word norm is much
larger than that of any g such that x(g) 6= 0 (see Definition 4.1).
• Show that the restriction of c to G × ∆(~0) is cohomologous to φ with transfer function b
(Proposition 4.3).
• Show that b(x) only depends on a bounded set of coordinates of x ∈ ∆(~0), and hence b may
be extended continuously to all of AG (Proposition 4.4).
• Show that c is cohomologous to φ with transfer function b (Theorem 4.5).
Acknowledgments. The author is grateful for the support of NSF grant 1502608. He also wishes
to thank Yongle Jiang for discussing his work [1], and Andy Putman for some helpful suggestions.

2

2

Geometry of groups

We now briefly survey some relevant basic facts about the geometry of finitely generated groups.
In particular, we recall the definitions of number of ends of a group and geodesics in a group, show
that biinfinite geodesics exist in any infinite group (Lemma 2.3), and show that for L > 0 and γ a
geodesic, the L-neighborhood of γ({0, 1, 2 . . .}) intersects the L-neighborhood of γ({0, −1, −2, . . .})
in a set contained inside the ball of radius 3L around γ(0) (Lemma 2.4).
Notation, assumptions, and conventions. For the rest of this paper, S will denote a finite,
symmetric (i.e., S = S −1 ) generating set for the infinite group G. Denote the identity element of G
by 1G . Every element of G is represented by some word in S, and the word norm |g| of any g ∈ G
is defined to be the length of the shortest word in S representing g. That is,
|g| := inf{ℓ : g = s1 . . . sℓ ; s1 , . . . , sℓ ∈ S}.
For g, h ∈ G, define the left-invariant distance
d(g, h) := |g −1 h|,
and for any T ⊆ G, let NL T := {g ∈ G : inf{d(g, s) : s ∈ T } ≤ L} denote the L-neighborhood of
T . Let B(r, g) denote the closed r-ball around g ∈ G, i.e., B(r, g) := Nr {g}. Many of our figures
are van Kampen diagrams—i.e., they depict planar graphs, with directed edges labeled by group
elements, such that every loop represents the identity.
Definition 2.1. Let I be an interval in Z—i.e., the intersection of Z with some possibly unbounded
real interval. A function γ : I→G is said to be a geodesic if d(γ(m), γ(n)) = |n−m| for all n, m ∈ I.
We say that γ : I→G is a path if d(γ(j − 1), γ(j)) = 1 whenever j − 1 and j are in I.
Obviously, this is equivalent to considering paths and geodesics in the Cayley graph of G. For
example, if Z2 is equipped with the standard generating set, then the map γ : Z → Z2 given by
γ : n 7→ (n, 0) is a geodesic.
Lemma 2.2. If g ∈ G, there exists a geodesic γ : {0, . . . , |g|}→G such that γ(0) = 1G and
γ(|g|) = g.
Proof. Let s1 . . . sℓ be a minimal length word in S ∗ representing g. By definition of the word metric,
ℓ = |g|. For j = 1, . . . , ℓ, set γ(j) to be the element of g represented by the length j prefix s1 . . . sj
of s1 . . . sℓ , and set γ(0) = 1G . Because s1 . . . sℓ has minimal length among words representing g,
sj+1 . . . sk has minimal length among words representing γ(j)−1 γ(k) for any 0 ≤ j < k ≤ ℓ. In
particular, |γ(j)−1 γ(k)| = k − j for all such j, k—i.e., γ is a geodesic.
Recall our assumption that G is infinite.
Lemma 2.3. There exists a geodesic γ : Z →G.
Proof. Because G is infinite, the word norm on G achieves every natural number. For each r ∈ N,
choose some gr ∈ G such that |gr | = 2r. By Lemma 2.2, there is a geodesic γr : {0, . . . , 2r}→G such
that γr (0) = 1G and γr (2r) = gr . Let γ
er : {−r, . . . , r}→G be defined by e
γr (n) := γr (r)−1 γr (n + r),
so that γ
er is a geodesic and γ
er (0) = 1G . By a trivial case of Arzela-Ascoli, the sequence (e
γr )r∈N
subconverges pointwise to a geodesic γ : Z →G.
3

g

≤L

≤L

γ(n+ )
γ(n− )
γ(−2L)

γ(2L)

γ(0)

Figure 2: A depiction of the proof of Lemma 2.4.

Let Z≥0 = {n ∈ Z : n ≥ 0} and Z≤0 := {n ∈ Z : n ≤ 0}. Recall that NL (T ) denotes the Lneighborhood of T ⊆ G. The following lemma bounds the intersection between the L-neighborhoods
of the left and right halves of a geodesic γ : Z →G. It may be compared to [1, Lemma 3.3], which
proves an analogous result for a path of the form n 7→ an (where a has infinite order).
Lemma 2.4. If γ : Z →G is a geodesic, then
NL (γ(Z≥0 )) ∩ NL (γ(Z≤0 )) ⊆ B(3L, 1G ).
(See Figure 2).
Proof. Suppose that g ∈ NL (γ(Z≥0 )) ∩ NL (γ(Z≤0 )). Then there exists n+ ≥ 0 and n− ≤ 0 such
that d(g, γ(n+ )) ≤ L and d(g, γ(n− )) ≤ L. Because γ is a geodesic,
|n+ − n− | = d(γ(n+ ), γ(n− )) ≤ d(γ(n+ ), g) + d(g, γ(n− )) ≤ 2L,
so n+ ≤ n− + |n+ − n− | ≤ 2L. Thus
|g| = d(g, 1G ) ≤ d(g, γ(n+ )) + d(γ(n+ ), 1G ) ≤ L + 2L = 3L

Ends of groups.

Let CayS G denote the Cayley graph of G with respect to S.

Definition 2.5. The number of ends of G is the limit, as n→∞, of the number of unbounded
connected components of CayS G \ B(n, 1G ).
Assumption: one endedness. For the rest of the paper, we will assume that G is one-ended.
This implies that for any r, CayS G \ B(r, 1G ) has exactly one unbounded connected component.
Definition 2.6. For r ∈ N, let
N (r) := sup{|g| : g does not belong to the unbounded component of CayS G \ B(r, 1G )}.
Note that N (r) < ∞, as, by bounded degree of CayS G, the supremum is taken over a finite
set—the number of bounded components is finite since each is adjacent to B(r, 1G ) in CayS G.

4

3

Cocycles over the full shift.

Recall that, by standing assumption, G is a one ended group. For the rest of the paper, fix a
continuous cocycle c : G × AG → H, where H is a discrete group. That c is continuous means that,
for every g ∈ G, there is some finite subset F ⊆ G (depending on g) such that c(g, x) depends only
on the restriction x|F of x to F . That is, if x|F = y|F then c(g, x) = c(g, y).
Definition 3.1. Let L ∈ N be such that, for any generator s ∈ S, the function c(s, x) is determined
by x|B(L,1G ) .
Recall that 0 denotes some element of A.
Definition 3.2. Let ~0 ∈ AG be the all-0 configuration, i.e., ~0(g) = 0 for all g ∈ G. Let ∆(~0) ⊆ AG
consist of all x ∈ AG such that x(g) = 0 for all but finitely many g ∈ G, and for x ∈ ∆(~0), let kxk
denote sup{|g| : x(g) 6= 0}.
Recall our definition of “path” from Definition 2.1.
Lemma 3.3. Let γ : {0, . . . , n}→G be path, and take x ∈ AG . Then c(γ(n)−1 , x)c(γ(0)−1 , x)−1 is
determined by x|NL (γ{0,...,n}) . That is, if
x|NL (γ{0,...,n}) = y|NL (γ{0,...,n}) ,
then
c(γ(n)−1 , x)c(γ(0)−1 , x)−1 = c(γ(n)−1 , y)c(γ(0)−1 , y)−1 .
(See Figure 3).
Proof. Let sj denote γ(j − 1)−1 γ(j) for j = 1, . . . , n. Since γ is a path, we have that sj ∈ S. From
the cocycle identity, we see that
−1
−1
c(γ(j)−1 , x) = c(s−1
j , γ(j − 1) x)c(γ(j − 1) , x).

Expanding recursively, we obtain
c(γ(n)−1 , x)
−1
−1
−1
= c(s−1
n , γ(n − 1) x)c(sn−1 , γ(n − 2) x)
−1
−1
. . . c(s−1
1 , γ(0) x)c(γ(0) , x).
−1
−1
Now, c(s−1
j , γ(j − 1) x) is determined by (γ(j − 1) x)|B(L,1G ) , which in turn is determined by
x|B(L,γ(j−1)) because (γ(j − 1)−1 x)(g) = x(γ(j − 1)g).

5

γ(0)−1 x
−1
c(s−1
x)
1 , γ(0)

c(γ(0)−1 , x)
γ(1)−1 x
γ(j − 1)

−1

x

c(γ(j − 1)−1 , x)
x

c(γ(1)−1 , x)

x

...

−1
c(s−1
x)
j , γ(j − 1)

c(γ(j)−1 , x)

c(γ(n − 1)−1 , x)

γ(j)−1 x
c(γ(n)−1 , x)

γ(n − 1)−1 x
−1
c(s−1
x)
n , γ(n − 1)

γ(n)−1 x

Figure 3: A pair of van Kampen diagrams depicting the proof of Lemma 3.3.

4

Proof of the main theorem.

Recall that, by standing assumption G is a one ended group, and recall the notations ~0, ∆(~0) and kxk
from Definition 3.2. We will now show that the continuous cocycle c : G × AG →H is cohomologous
to a homomorphism—i.e., that there exists a homomorphism φ : G→H and a continuous function
b : AG →H such that c(g, x) = b(gx)φ(g)b(x)−1 . As a first step towards finding such b and φ, observe
that if we had such b and φ, then we would have c(g, ~0) = b(~0)φ(g)b(~0)−1 for every g ∈ G. For any
x ∈ ∆(~0) and |g| sufficiently large depending on x, we would have b(gx) = b(~0) by continuity of
b, and hence c(g, x) = b(~0)φ(g)b(x)−1 . Equivalently, b(x) would equal b(~0)φ(g)c(g, x)−1 . Thus, the
following definition of φ and b|∆(~0) corresponds to the normalization b(~0) = 1H .
Definition 4.1. Let φ(g) := c(g, ~0). For each x ∈ ∆(~0), fix some gx ∈ G such that |gx | >
N (kxk + L) and set b(x) := c(gx , x)−1 φ(gx ).
Observe that φ is a homomorphism by the cocycle identity, since ~0 is a fixed point for the action
of G on AG . We will show that the restriction of c to G × ∆(~0) is cohomologous to φ with transfer
function b, and then we will show that b extends continuously to all of AG . First, we see that the
choice of gx in Definition 4.1 is irrelevant.
Lemma 4.2. If |g| > N (kxk + L) then c(g, x)−1 φ(g) = b(x).
Proof. Let γ : {0, . . . , n} be a path such that γ connects g−1 to gx−1 outside of B(kxk + L, 1G ) (so
that γ(0)−1 = g and γ(n)−1 = gx and |γ(j)| > kxk + L for j = 0, . . . , n). Such a path exists because
|g| and |gx | are greater than N (kxk + L). Since |γ(j)| > kxk + L for all j ∈ {0, . . . , n} and x(g) = 0
whenever |g| > kxk, we have that x(g) = 0 for any g ∈ NL (γ{0, . . . , n}), so by Lemma 3.3, we have
6

c(γ(n)−1 , x)c(γ(0)−1 , x)−1 = c(γ(n)−1 , ~0)c(γ(0)−1 , ~0)−1 .
Consequently
c(gx , x)c(g, x)−1 = φ(gx g−1 ),
or equivalently
c(g, x)−1 φ(g) = c(gx , x)−1 φ(gx ) = b(x)
as desired.
We now show that the restriction of c to G × ∆(~0) is cohomologous to φ with transfer function
b.
Proposition 4.3. For all g ∈ G and x ∈ ∆(~0), we have c(g, x) = b(gx)φ(g)b(x)−1 .
(See Figure 4).
Proof. Choose some e
g ∈ G such that |e
g | > N (kxk + L) and |e
g | > |g| + N (kgxk + L). Then we have
|e
g g −1 | ≥ |e
g | − |g| > N (kgxk + L), so by Lemma 4.2 we have b(gx) = c(e
g g−1 , gx)−1 φ(e
g g−1 ) and
−1
−1
−1
−1
−1
b(x) = c(e
g , x) φ(e
g ). It follows that c(e
g g , gx) = b(gx)φ(e
g g ) and c(e
g , x) = φ(e
g )b(x)−1 .
By the cocycle identity,
c(e
g , x) = c(e
g g−1 , gx)c(g, x).
Rearranging, we obtain
c(g, x) = c(e
g g −1 , gx)−1 c(e
g , x) = (b(gx)φ(e
g g−1 )−1 )(φ(e
g )b(x)−1 )
= b(gx)φ(g)b(x)−1
as desired.
As ∆(~0) is noncompact, it is not even obvious a priori that b takes on only finitely many values.
However, we now show that x 7→ b(x) factors through the restriction x 7→ x|B(3L,1G ) , so that b may
be extended continuously to a function AG →H. The following proposition should be compared to
[1, Lemmas 3.5,4.1]
Proposition 4.4. If x, y ∈ ∆(~0) with x|B(3L,1 ) = y|B(3L,1 ) then b(x) = b(y).
G

G

Proof. By Lemma 2.3 there exists a geodesic γ : Z →G. We will begin by finding z ∈ ∆(~0) such
that z|NL (γ(Z≥0 )) = x|NL (γ(Z≥0 )) and z|NL (γ(Z≤0 )) = y|NL (γ(Z≤0 )) .
Let z(g) := 0 for g ∈
/ NL (γ(Z≥0 )) ∪ NL (γ(Z≤0 )), and take z(g) := x(g) for g ∈ NL (γ(Z≥0 ))
and z(g) := y(g) for g ∈ NL (γ(Z≤0 )). For g ∈ NL (γ(Z≤0 )) ∩ NL (γ(Z≥0 )) ⊆ B(3L, 1G ) (recall
Lemma 2.4), these definitions agree by our assumption that x and y agree on B(3L, 1G ). Because
x, y ∈ ∆(~0), we also have z ∈ ∆(~0).
By Lemma 3.3, c(γ(n)−1 , x) = c(γ(n)−1 , z) for all n ≥ 0. By Lemma 4.2, this implies that
b(x) = b(z). Arguing similarly with γ(−n) in place of γ(n), we see that b(y) = b(z). It follows that
b(x) = b(y).
We will now prove our main theorem.
Theorem 4.5. Every continuous cocycle c : G × AG →H is cohomologous to a homomorphism.
Proof. Take b and φ as given in Definition 4.1. We will first note that b extends continuously to all
of AG , and then we will show that this extension is a transfer function for a cohomology from c to
φ.
7

c(g, x)
gx

b(gx)

A

φ(g)

b(x)
x

φ(e
g g −1 )

c(e
g g −1 , gx)

B

φ(e
g)

1G

C
c(e
g , x)

gx
e

Figure 4: A van Kampen diagram depicting the proof of Proposition 4.3. The inner and outer
circles are relations by the cocycle identity. B and C are relations by Lemma 4.2. It follows that A
is a relation.
Extending b over AG . For x ∈ AG , let x ∈ ∆(~0) be given by setting x(g) := x(g) for g ∈
B(3L, 1G ) and x(g) := 0 for g ∈
/ B(3L, 1G ). For arbitrary x ∈ AG we may now define b(x) := b(x).
By definition b is continuous, and by Proposition 4.4, we have that b : AG →H agrees with Definition
4.1 on ∆(~0). We now proceed to show that c(g, x) = b(gx)φ(g)b(x)−1 for every g ∈ G and x ∈ AG .
Showing that c is cohomologous to φ with transfer function b. Given g ∈ G and x ∈ AG ,
we shall apply Proposition 4.3 to some x
e ∈ ∆(~0) approximating x. Let x
e ∈ ∆(~0) be given by
′
′
′
′
′
setting x
e(g ) := x(g ) for g ∈ B(|g| + 3L, 1G ) and x
e(g ) := 0 for g ∈
/ B(|g| + 3L, 1G ). By Lemma
2.2, there is a geodesic γ : {0, . . . , n}→G such that γ(0) = 1G and γ(n) = g−1 . For such a
geodesic, NL (γ{0, . . . , n}) ⊆ B(|g| + 3L, 1G ). It follows by Lemma 3.3 that c(g, x)c(1G , x)−1 =
c(g, x
e)c(1G , x
e)−1 , or in other words
c(g, x) = c(g, x
e).
We have c(g, x
e) = b(ge
x)φ(g)b(e
x )−1 by Proposition 4.3, as x
e ∈ ∆(~0).
We have b(e
x) = b(x) = b(x) by definition. We see that
(ge
x)|B(3L,1G ) = (gx)|B(3L,1G )
because g−1 B(3L, 1G ) ⊆ B(|g| + 3L, 1G ). Thus by Proposition 4.4, b(ge
x) = b(gx). It follows that
c(g, x) = c(g, x
e) = b(ge
x)φ(g)b(e
x)−1 = b(gx)φ(g)b(x)−1 ,
as desired.

8

References
[1] Nhan-Phu Chung and Yongle Jiang. Continuous cocycle superrigidity for shifts and groups with
one end. Mathematische Annalen, pages 1–24, 2016.
[2] Xin Li. Continuous orbit equivalence rigidity. Ergodic Theory and Dynamical Systems, pages
1–21, 2016.
[3] Xin Li. Dynamic characterizations of quasi-isometry, and applications to cohomology. arXiv
preprint arXiv:1604.07375, 2016.

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