[Paper Review] A non-LEA Sofic Group

Aditi Kar, Nikolay Nikolov|arXiv (Cornell University)|May 2, 2014

Geometric and Algebraic Topology5 references4 citations

TL;DR

This paper constructs the first known example of a finitely presented sofic group that is not LEA (locally embeddable in amenable groups), demonstrating that the class of LEA groups is not closed under free products with amalgamation over infinite cyclic subgroups. The authors prove that the amalgamated free product $ G = \mathrm{SL}_n(\mathbb{Z}[1/p]) \ast_Z \mathrm{SL}_n(\mathbb{Z}[1/p]) $, where $ Z $ is an infinite cyclic subgroup, is sofic but not residually amenable, and hence not LEA, by showing it lacks a co-amenable LEA subgroup.

ABSTRACT

We describe elementary examples of finitely presented sofic groups which are not residually amenable (and thus not initially subamenable or LEA, for short). We ask if an amalgam of two amenable groups over a finite subgroup is residually amenable and answer this positively for some special cases, including countable locally finite groups, residually nilpotent groups and others.

Motivation & Objective

• To construct explicit examples of finitely presented sofic groups that are not residually amenable (and thus not LEA).
• To investigate whether free products with amalgamation over finite subgroups preserve residual amenability.
• To determine conditions under which $ A \ast_C B $ is residually amenable when $ A $ and $ B $ are residually amenable and $ C $ is finite.
• To clarify the relationship between co-amenability, LEA subgroups, and the structure of amalgamated free products in sofic groups.

Proposed method

• Construct the group $ G = \mathrm{SL}_n(\mathbb{Z}[1/p]) \ast_Z \mathrm{SL}_n(\mathbb{Z}[1/p]) $, where $ Z $ is the infinite cyclic subgroup generated by a unipotent matrix.
• Use the soficity of amalgamated free products over amenable subgroups, established in prior work, to confirm $ G $ is sofic.
• Assume for contradiction that $ G $ has a co-amenable LEA subgroup $ H $, and analyze the action of $ G $ on the coset space $ X = G/H $ via a $ G $-invariant mean.
• Leverage property (T) of $ \mathrm{SL}_n(\mathbb{Z}[1/p]) $ to show that the stabilizer of a generic point in $ X $ must have finite index in each factor, leading to a contradiction in the image of $ H $ under homomorphisms to amenable groups.
• Use the structure of divisible subgroups $ Y_i \cong \mathbb{Z}[1/p] $ and their intersections with $ H $ to construct elements $ \beta_1, \beta_2 \in H $ whose images in any amenable group must be equal despite being distinct in $ H $, violating the existence of a partial monomorphism.
• Apply results on the derived series and profinite topologies to establish residual amenability in special cases, such as when $ C $ is disjoint from lower central series terms of $ A $ and $ B $.

Experimental results

Research questions

• RQ1 Is there a finitely presented sofic group that is not LEA?
• RQ2 Does the class of residually amenable groups remain closed under free products with amalgamation over finite subgroups?
• RQ3 Can a sofic group with a co-amenable LEA subgroup exist if it contains a copy of $ \mathrm{SL}_n(\mathbb{Z}[1/p]) $ for $ n \geq 3 $?
• RQ4 Under what conditions is $ A \ast_C B $ residually amenable when $ A $ and $ B $ are residually amenable and $ C $ is finite?
• RQ5 Is the fundamental group of a graph of finite groups with bounded vertex groups virtually free?

Key findings

• The group $ G = \mathrm{SL}_n(\mathbb{Z}[1/p]) \ast_Z \mathrm{SL}_n(\mathbb{Z}[1/p]) $ is sofic, as guaranteed by the theorem on amalgamated free products of sofic groups over amenable subgroups.
• The group $ G $ is not LEA, as it does not admit a co-amenable LEA subgroup.
• Any co-amenable subgroup of $ G $ must intersect each factor $ \mathrm{SL}_n(\mathbb{Z}[1/p]) $ in a finite-index subgroup.
• The images of the elements $ \beta_1 = f_1(y(a/p)) $ and $ \beta_2 = f_2(y(a/p)) $ in any homomorphism to an amenable group must be equal, despite $ \beta_1 \neq \beta_2 $ in $ G $, which contradicts the existence of a partial monomorphism to an amenable group.
• The amalgamated free product $ A \ast_C B $ is residually amenable when $ C $ is finite and disjoint from the lower central series of $ A $ and $ B $, as shown in Corollary 8.
• When $ C $ is Hausdorff in the profinite topologies of $ A $ and $ B $, the amalgam $ A \ast_C B $ is residually amenable, as established in Theorem 7.