[Paper Review] Sofic groups and profinite topology on free groups

Lev Glebsky, Luis Manuel Rivera Martinez|ArXiv.org|Sep 1, 2007

Geometric and Algebraic Topology3 citations

TL;DR

This paper introduces weakly sofic groups (w-sofic groups) as a generalization of sofic groups, defined via bi-invariant metrics on finite groups rather than the Hamming metric on symmetric groups. The key contribution is establishing that the existence of non-w-sofic groups is equivalent to a profinite topology property: the closure of products of conjugacy classes in a free group not being contained in the normal subgroup they generate. This links group theory, profinite topology, and the broader conjecture on soficity.

ABSTRACT

We give a definition of weakly sofic groups (w-sofic groups). Our definition is rather natural extension of the definition of sofic groups where instead of Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non w-sofic groups is equivalent to some conjecture about profinite topology on free groups.

Motivation & Objective

• To define and study weakly sofic groups (w-sofic groups), a generalization of sofic groups using arbitrary bi-invariant metrics on finite groups instead of the Hamming metric on symmetric groups.
• To investigate the connection between w-sofic groups and the profinite topology on finitely generated free groups.
• To explore whether products of conjugacy classes in free groups are closed in the profinite topology, as this determines the existence of non-w-sofic groups.
• To establish that the non-closure of such products in the profinite topology is equivalent to the existence of non-w-sofic groups.
• To provide a characterization of finite separability of normal subgroups in free groups in terms of conjugacy class products and profinite closure.

Proposed method

• Define w-sofic groups via $(\Phi,\epsilon,\alpha)$-homomorphisms into finite groups equipped with bi-invariant metrics, generalizing the sofic group definition.
• Use the profinite topology on a finitely generated free group $F$, where a set is closed if it is separable via homomorphisms to finite groups.
• Prove that a normal subgroup $N \triangleleft F$ is finitely separable if and only if the closure of any product of conjugacy classes $[g_1]^F \cdots [g_k]^F$ for $g_i \in N$ is contained in $N$.
• Employ non-standard analysis techniques by considering hyperfinite groups with internal bi-invariant metrics and their infinitesimal neighborhoods to characterize w-soficity.
• Use the fact that $F/N$ is w-sofic if and only if there exists a homomorphism $\phi: F \to H$ with $\phi(N) \subseteq \mathcal{E}$ and $\phi(F \setminus N) \cap \mathcal{E} = \emptyset$, where $\mathcal{E}$ is the set of elements with infinitesimal distance to identity in $H$.
• Apply the characterization of profinite closure via homomorphisms to finite groups: $x \in \overline{X}$ iff $\phi(x) \in \phi(X)$ for all such $\phi$.

Experimental results

Research questions

• RQ1 Is there a non-w-sofic group, and if so, what does this imply about the profinite topology on free groups?
• RQ2 Are products of conjugacy classes in a free group closed in the profinite topology?
• RQ3 Is the closure of a product of conjugacy classes in a free group contained in the normal subgroup they generate?
• RQ4 Can the finite separability of normal subgroups in free groups be characterized via conjugacy class products and profinite closure?
• RQ5 Does the existence of a non-w-sofic group imply a failure of the profinite topology to close products of conjugacy classes?

Key findings

• The existence of a non-w-sofic group is equivalent to the existence of a finitely generated free group $F$ and elements $g_1, \dots, g_k \in F$ such that $\overline{[g_1]^F \cdots [g_k]^F} \not\subseteq N(g_1, \dots, g_k)$, where $N(g_1, \dots, g_k)$ is the normal subgroup generated by the $g_i$.
• A normal subgroup $N \triangleleft F$ is finitely separable if and only if for all $g_1, \dots, g_k \in N$, the closure of $[g_1]^F \cdots [g_k]^F$ in the profinite topology is contained in $N$.
• The paper shows that in the pro-$p$ topology, there exist elements $g_1, g_2$ in the 2-generated free group such that $[g_1]^F [g_2]^F$ is not closed, and its closure contains an element not in $N(g_1, g_2)$, suggesting a similar failure may occur in the profinite topology.
• The paper proves that $F/N$ is w-sofic if and only if there exists a homomorphism $\phi: F \to H$ to a finite group $H$ such that $\phi(N) \subseteq \mathcal{E}$ and $\phi(F \setminus N) \cap \mathcal{E} = \emptyset$, where $\mathcal{E}$ is the infinitesimal neighborhood of the identity in a hyperfinite group $H$.
• The characterization of profinite closure via homomorphisms to finite groups is used to show that $x \in \overline{X}$ iff $\phi(x) \in \phi(X)$ for all homomorphisms $\phi$ to finite groups.
• The paper establishes that the standard definition of soficity (using Hamming metric on $S_n$) is a special case of w-soficity, and thus w-sofic groups form a strictly larger class than sofic groups.