Group cubization

 It is a highly non-trivial fact that, given a free group $F_m$ on $m \geq 2$ generators, one can add the relations $w^n$ for all possible $w \in F_m$ and obtain, for sufficiently large $n$, a group which isn't finitely presented, in particular an infinite group. These are called the free Burnside groups, denoted $B(m,n)$. It was open for a long time whether there exist finitely generated infinite groups of bounded exponent. Many people, myself included, consider free Burnside groups to be a class of pretty interesting monster groups. What is equally interesting is that there is nothing special about the free group: this works with any torsion-free non-elementary hyperbolic group. The interested reader should consult Coulon's papers for details; all methods of showing these are infinite revolve around small cancellation methods. If one starts with such a hyperbolic group with Kazhdan's property (T), then one obtains finitely generated infinite groups of bounded exponent which additionally have (T).

Shalom asked whether the infinite free Burnside groups themselves have Property (T), which is a reasonable enough question. In this post I will outline two approaches, both with their own merits, to this question, due to Osajda and Pichot respectively. 

Theorem 1: If $B(m,n)$ is infinite then, for any integer $k>1$, $B(m,kn)$ acts on a $\mathrm{CAT}(0)$ cube complex with unbounded orbits, so doesn't have (T).
(the statement Pichot proves is slightly different but morally the same)

Group cubization

Let $\Gamma=\mathrm{Cay(G,S)}$ be a Cayley graph of a group $G$ generated by a finite symmetric set $S$ (that is, $S=S^{-1}$).
We use the convention that every vertex belongs to two edges corresponding to a pair $\{s,s^{-1}\}$, for each $s\in S$. In particular, generators being involutions give rise to double edges, and the degree of vertices is $|S|$.  Fix an integer $k>1$. Let $p\colon \widetilde{\Gamma} \to \Gamma$ be the \emph{$\mathbb \mathbb{Z}_k$--homology covering} of $\Gamma$, that is, the covering corresponding to the kernel $K$ of the natural map $\pi_1(\Gamma,1)\to H_1(\Gamma;\mathbb \mathbb{Z}_k)=
\bigoplus_I \mathbb \mathbb{Z}_k$, where $I$ is the set indexing generators of $\pi_1(\Gamma,1)$, and $1$ is the vertex of $\Gamma$ being the identity of $G$.
Observe that it is a characteristic covering, that is, $K$ is a characteristic subgroup of $\pi_1(\Gamma,1)$. This subgroup can be identified with $\pi_1(\widetilde{\Gamma},\widetilde{1})$, where  $\widetilde{1}$ is a vertex with $p(\widetilde{1})=1$. 
Therefore, for any automorphism $g$ of $\Gamma$, we have 
$g_{\ast}\circ p_{\ast} (\pi_1(\widetilde{\Gamma},\widetilde{1}))=p_{\ast} (\pi_1(\widetilde{\Gamma},\widetilde{1}))$, and hence
$g\circ p\colon \widetilde{\Gamma} \to \Gamma$ can be lifted to a map $\widetilde{G} \colon \widetilde{\Gamma} \to \widetilde{\Gamma}$ satisfying
$p\circ \widetilde{G}= g \circ p$.

Every element $g\in G$ defines an automorphism of $\Gamma$ given by left multiplication by $g$ (and also denoted by $g$). Let $\widetilde{G}$ denote the set of all lifts of all such automorphisms. This turns out to have a well-described Cayley graph:

Theorem 2: The set $\widetilde{G}$ forms a group generated by a set $\widetilde{S}$ with $|\widetilde{S}|=|S|$, and with $\widetilde{\Gamma}$ being its Cayley graph $\mathrm{Cay}(\widetilde{G},\widetilde{S})$. The group $\widetilde{G}$ is called the cubization of a group $G$ with respect to its Cayley graph $\Gamma=\mathrm{Cay(G,S)}$.

The preimage of every open edge of $\Gamma$ disconnects the cover $\widetilde{\Gamma}$ into $k$ connected components.
Such a partition of vertices of $\widetilde{\Gamma}$ gives rise to $2^{k-1}-1$ partitions
into two nonempty sets -- these are walls in the space with walls  $(\widetilde{\Gamma}^{(0)}, \mathcal W)$.
The group $\widetilde{G}$ acts on $(\widetilde{\Gamma}^{(0)}, \mathcal W)$: for every generator $\widetilde{S} \in \widetilde{S}$ a wall corresponding to $e\in E$ is mapped by
$\widetilde{S}$ to a wall corresponding to $s(e)$. Clearly, the action has orbits that are unbounded
with respect to the wall pseudo-metric. 

In the case when there exists an edge separating $\Gamma$, the translates (by $G$) of this edge
define the space with walls $(\Gamma^{(0)},\mathcal W')$. The $G$--action on $(\Gamma^{(0)},\mathcal W')$ has unbounded orbits, and induces a $\widetilde{G}$--action on $(\Gamma^{(0)},\mathcal W')$ with the same property. Existing results in the literature then imply that $\widetilde{G}$ acts with unbounded orbits on a corresponding CAT(0) cubical complex, which is morally similar to how a wallspace gives rise to a cubulation as discussed in a previous blog post. This proves

Lemma 3: If a group defined by a presentation $\langle S \; | \; r_1,r_2,\ldots \rangle$ is infinite
    then the group with the presentation $\langle S \; | \; r_1^k,r_2^k,\ldots \rangle$ acts with unbounded
    orbits on a CAT(0) cube complex.

I find this quite appealing: it isn't very often that one sees such clear geometry almost purely from a presentation. Theorem 1 follows quickly from this and the slick result of Niblo--Reeves that we now discuss

Property (T) vs cube complexes

Definition: A conditionally negative kernel on a set \( V \) is a function \( f: V \times V \to \mathbb{R} \) such that for any finite subset \( \{v_1, \ldots, v_n\} \subset V \) and any real numbers \( \{\lambda_1, \ldots, \lambda_n\} \) such that 
\[
\sum_i \lambda_i = 0
\]
the following inequality holds:
\[
\sum_{i,j} \lambda_i \lambda_j f(v_i, v_j) \leq 0.
\]

A conditionally negative kernel on a group \( G \) is a conditionally negative kernel on the set of elements of \( G \) such that for any \( g, h, k \) in \( G \), \( f(gh, gk) = f(h, k) \). It is a theorem that a finitely generated group \( G \) has Kazhdan's property (T) (or is a \( T \)-group) if and only if every conditionally negative kernel on \( G \) is bounded. We will use this characterisation to prove

Theorem 4: If \( G \) is a group satisfying Kazhdan's property (T) and \( X \) is a finite dimensional CAT(0) cube complex on which \( G \) acts cellularly (and therefore isometrically), then the action has a global fixed point.

The set of vertices of the cubing \( X \) can be viewed as a discrete metric space, where the metric \( D(u,v) \) is given by the length of the shortest edge path between the vertices \( u \) and \( v \) in the 1-skeleton of \( X \). If \( X \) is finite dimensional this metric is quasi-isometric to the CAT(0) metric (it is at most \( \sqrt{n} \) times the CAT(0) metric, where \( n \) is the dimension of the complex; this does not require the complex to be cocompact). The shortest path in the 1-skeleton crosses any hyperplane at most once, and since every edge crosses exactly one hyperplane, the distance between two vertices is the number of hyperplanes separating them. Recalling that a hyperplane separates the vertices into two half spaces \( U^+ \) and \( U^- \), it follows that
\[
D(u,v) = \sum_U \chi_U(u)(1 - \chi_U(v))
\]
where \( \chi_U \) is the characteristic function of the half space \( U \), and \( U \) ranges over all the half spaces in \( X \). Although there are infinitely many half spaces, the fact that only finitely many of them separate any pair of vertices means that this will be a finite sum.

Now suppose that \( G \) is a group acting cellularly on \( X \). The action restricts to the 1-skeleton and therefore preserves the metric \( D \), so we have established parts 2 and 3 of the following lemma:

Lemma 5: For any CAT(0) cube complex \( X \) the simplicial metric \( D \) on the 1-skeleton of \( X \) satisfies the following properties:

1. \( D \) is a conditionally negative kernel on the vertex set of \( X \).
2. \( D \) is invariant under the action of \( G \) on the vertex set.
3. For any vertices \( u, v \in X \), \( d(u, v) \leq D(u, v) \leq \sqrt{n} \, d(u, v) \) where \( n \) is the dimension of the cube complex.

Proof: It remains to establish that the metric \( D \) is a conditionally negative kernel. Let \( u_1, u_2, \ldots, u_n \in G \) and \( \lambda_i \in \mathbb{R} \) with \( \sum_i \lambda_i = 0 \). Then
\[
\sum_i \sum_j \lambda_i \lambda_j D(u_i, u_j) = \sum_i \sum_j \lambda_i \lambda_j \sum_U \chi_U(u_i) (1 - \chi_U(u_j))
\]
where the third sum can be taken over the finitely many half spaces \( U \) separating the vertices \( \{u_1, \ldots, u_n\} \) (the other half spaces all contribute 0 to the sum). This triple sum can now be rearranged to get
\[
\sum_i \sum_j \lambda_i \lambda_j \sum_U \chi_U(u_i) - \sum_i \sum_j \lambda_i \lambda_j \sum_U \chi_U(u_i)\chi_U(u_j).
\]
The first term is zero since it can be rearranged to give \( \sum_j \lambda_j \sum_i \lambda_i \sum_U \chi_U(u_i) \), and \( \sum_j \lambda_j = 0 \) by hypothesis. Likewise the second term can be rewritten as
\[
\sum_U \sum_i \lambda_i \chi_U(u_i) \sum_j \lambda_j \chi_U(u_j) = \sum_U \left( \sum_i \lambda_i \chi_U(u_i) \right)^2
\]
which is positive. It follows that the entire sum is negative as required. $\blacksquare$

We now have all the ingredients we need for the

Proof of theorem 4: For any CAT(0) cube complex \( X \), the simplicial metric \( D \) on the 1-skeleton of \( X \) restricts to a conditionally negative kernel on the vertex set \( X^{(0)} \). Since \( G \) acts cellularly on \( X \), \( D \) is invariant under the action, so setting \( f(g, h) = D(gv, hv) \) gives a conditionally negative kernel on the group \( G \). Since \( G \) has Kazhdan's property (T) the conditionally negative kernel must be bounded so the orbit \( Gv \) is bounded in the metric \( D \) and therefore, since \( X \) is finite dimensional, in the metric \( d \) (see section 3). Since \( X \) is finite dimensional it is complete and in a complete CAT(0) metric space any isometric action with a bounded orbit has a global fixed point. $\blacksquare$

Wreath products

Pichot's approach is based on wreath products, which we now recall.

Let $G$ be a group, and $X$ a $G$-set. Let $W$ be another group. Then $G$ acts on the
direct sum $\oplus _X W$ by permutations of factors. The (permutational) wreath product
$W \wr_X G$ is defined to be the semidirect product $W ^{(X)} ⋊ G$. When the action of $G$ on $X$ is simply transitive, it is called the standard wreath product (this special case is
sometimes called the wreath product) and denoted by$ W \wr G$. Wreath products provide examples of exotic groups: for instance de Cornulier, extending a result of Baumslag, proved

Theorem 6: If $W \neq  \{1\}$, the wreath product $W \wr_X G$ is finitely presented if and only if the following conditions are satisfied

1.  $W$ and $G$ are finitely presented, 
2. $G$ acts on $X$ with finitely generated stabilisers, and
3. the product action of $G$ on the cartesian square $X^2$ has finitely many orbits.

de Cornulier uses this to obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroups. Another pathological example: $\mathbb{Z}  \wr \mathbb{Z}$ is an example of a RFRS group with infinite dimensional second homology. 

It is natural to wonder about other properties of groups, such as Property (T). (T) falls into a much larger framework, as we now see.

For a category $S$ of metric spaces, say that a group has property $BS$ if every action on a space in $S$ (preserving the structure) has a bounded orbit. Many well-studied properties fall under this framework: Bergman's property (all metric spaces), property FH (affine real Hilbert spaces), property FW (CAT(0) cube complexes), property FA (simplicial trees), property FR (real trees) and uncountable cofinality (ultrametric spaces).
Leemann--Schneeberger characterised when a wreath product has any of the above properties. Broadly speaking, they show theorems of the following form:
Let $S$ be one of the above properties. If $W$ is non-trivial, then $W \wr_X G$ has property $BS$ if and only if both $G$ and $W$ have property $BS$ and $X$ is finite. The precise statement depends on S, and can be more complicated than this.

While it was known for some of the properties before, the advantage of their approach is that it is really unified and gets to the heart, in some sense, of how these properties behave under the wreath product construction.

Using this, we can now give Pichot's one paragraph proof of a variant of theorem 1:
Consider the  wreath product $W:=\mathbb{Z}/k\mathbb{Z} \wr B(m,n)$ of $B(m,n)$ with a finite cyclic group. Notice that the order of every element in $W$ divides $kn$, and that $W$ is generated by $m+1$ elements. Therefore, by the universal property, $W$ is a quotient of the free Burnside group $B(m+1,kn)$ of exponent $kn$. The wreath product $H\wr G$ does not have Kazhdan's property (T) if $G$ is infinite and $H$ is not trivial.  
In particular, the group $B(m+1,kn)$ does not have Kazhdan's property (T) if $B(m,n)$ is infinite.

Finally, we remark that ideas about using the infiniteness of one Burnside group to deduce something about another group have appeared elsewhere: for instance in Bridson--Vogtmann's paper "Homomorphisms from automorphism groups of free groups".