[Paper Review] Determining Fuchsian groups by their finite quotients

Martin R. Bridson, Marston Conder+1 more|arXiv (Cornell University)|Jan 15, 2014

Geometric and Algebraic Topology27 references4 citations

TL;DR

This paper establishes that certain classes of finitely generated, residually finite groups—particularly Fuchsian groups, surface groups, right-angled Artin groups, and non-uniform arithmetic lattices—are uniquely determined up to isomorphism by their finite quotients. Using profinite group theory, the authors prove that if two such groups have identical sets of finite quotient isomorphism classes, then they are isomorphic, resolving a long-standing question in geometric group theory for key classes of groups.

ABSTRACT

Let $\C(Γ)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $Γ$. We investigate the extent to which $\C(Γ)$ determines $Γ$ when $Γ$ is a group of geometric interest. If $Γ_1$ is a lattice in ${ m{PSL}}(2,\R)$ and $Γ_2$ is a lattice in any connected Lie group, then $\C(Γ_1) = \C(Γ_2)$ implies that $Γ_1$ is isomorphic to $Γ_2$. If $F$ is a free group and $Γ$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(Γ)$ implies that $F\congΓ$. If $Γ_1

Motivation & Objective

• To determine whether the set of finite quotients of a group uniquely identifies it up to isomorphism within key classes of geometric groups.
• To extend the understanding of group classification via finite quotients beyond residual finiteness, focusing on Fuchsian groups and related classes.
• To resolve open questions about whether free groups, surface groups, and arithmetic lattices are distinguishable by their finite quotient sets.
• To provide a theoretical framework using profinite group theory to analyze group isomorphism invariants derived from finite quotients.
• To demonstrate that finite quotients can distinguish between non-isomorphic groups even when subgroup growth is asymptotically equivalent.

Proposed method

• The authors use profinite group theory to analyze the structure of the profinite completion of a group, which encodes all its finite quotients.
• They apply results from the theory of profinite rigidity, particularly the use of profinite invariants such as the set of finite quotients and lcm of generators.
• The proof relies on analyzing the structure of finite quotients via number-theoretic and group-theoretic constraints, including divisibility and lcm conditions on generators.
• Explicit constructions and case analysis are used to rule out isomorphisms between non-isomorphic triangle groups by exhibiting distinguishing finite quotients.
• The authors use algebraic number theory and properties of lattices in Lie groups to analyze non-uniform arithmetic lattices and their profinite completions.
• They apply results from Serre’s theory of good groups and LERF properties to distinguish surface groups and residually free groups from free groups.

Experimental results

Research questions

• RQ1 Can a Fuchsian group be uniquely determined by its set of finite quotients among all finitely generated, residually finite groups?
• RQ2 Do right-angled Artin groups and residually free groups with specific properties have unique profinite completions?
• RQ3 Can non-uniform arithmetic lattices in semisimple Lie groups be distinguished by their finite quotient sets?
• RQ4 Is there a finite quotient that distinguishes two non-isomorphic triangle groups?
• RQ5 To what extent do finite quotients determine the ambient Lie group structure of a lattice?

Key findings

• If Γ₁ is a Fuchsian group and Γ₂ is a lattice in a connected Lie group, then C(Γ₁) = C(Γ₂) implies Γ₁ ≅ Γ₂.
• For a free group F and a group Γ in certain classes (e.g., surface groups, residually free groups with a surface subgroup), C(F) = C(Γ) implies F ≅ Γ.
• For non-uniform arithmetic lattices in semisimple Lie groups with trivial center and no compact factors, C(Γ₁) = C(Γ₂) implies G ≅ PSL(2,C) and Γ₂ lies in one of finitely many commensurability classes.
• The paper provides an explicit construction of finite quotients that distinguish between two non-isomorphic triangle groups, using lcm and sum conditions on generators.
• The authors show that groups with asymptotically equivalent subgroup growth (in the sense of Müller and Schlage-Puchta) are distinguished by their finite quotients.
• A pair of non-isomorphic triples (r,s,t) and (u,v,w) with equal products and lcm values is ruled out via Corollary 8.10 and Lemma 8.11, confirming uniqueness in a key case.