Commensurability invariants for nonuniform tree lattices

On commensurability of right-angled Artin groups I: RAAGs defined by trees of diameter 4

In this paper we study the classification of right-angled Artin groups up to commensurability. We characterise the commensurability classes of RAAGs defined by trees of diameter 4. In particular, we prove a conjecture of Behrstock and Neumann that there are infinitely many commensurability classes. Hence, we give first examples of RAAGs that are quasi-isometric but not commensurable.

On commensurability of some right-angled Artin groups II: RAAGs defined by paths

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length $n>4$ is commensurable to a RAAG defined by a tree of diameter 4 if and only if $n$ is 2 modulo 4. These results follow from the connection that we establish between the classification of RAAGs up to commensurability and line…

Commensurability growth of branch groups

Fixing a subgroup $Γ$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ of $G$ with $[Γ: Γ\cap Δ][Δ: Γ\cap Δ] = n$. For pairs $Γ\leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Γ$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Γ$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $\aleph_0$ for any $n …

关于Leighton图覆盖定理的证明、与可公度性对偶的概念及正规虚拟收缩子

On the proofs of Leighton's Graph Covering Theorem, a notion dual to commensurability, and normal virtual retracts

本文通过反例说明Leighton定理的树格证明不充分，给出可公度化由共可公度化诱导的充要条件，并证明正规虚拟收缩子是虚拟直和项，回答Merladet和Minasyan的问题。

Erratum to the paper "Acylindrical hyperbolicity of groups acting on trees"

In our paper arXiv:1310.6289, we stated that acylindrical hyperbolicity of a group is invariant under commensurability up to finite kernels. Unfortunately, the proof of this fact contained a gap. The goal of this erratum is to point out the gap and to correct the statements and proofs of results of arXiv:1310.6289 and the subsequent paper arXiv:1501.03066 affected by it.

Lattices acting on right-angled buildings

Let X be a right-angled building. We show that the lattices in Aut(X) share many properties with tree lattices. For example, we characterise the set of covolumes of uniform and of nonuniform lattices in Aut(X), and show that the group Aut(X) admits an infinite ascending tower of uniform and of nonuniform lattices. These results are proved by constructing a functor from graphs of groups to complexes of groups.