[Paper Review] On Tarski's problem for virtually free groups

Simon André|arXiv (Cornell University)|Oct 18, 2019

Geometric and Algebraic Topology33 references4 citations

TL;DR

This paper provides a complete classification of finitely generated virtually free groups up to ∀∃-elementary equivalence, establishing that two such groups have the same ∀∃-theory if and only if they admit isomorphic multiple legal extensions via specific HNN extensions or subgroup replacements. The result yields a decidable algorithm to determine ∀∃-elementary equivalence from finite presentations, leveraging hyperbolic group isomorphism algorithms and structural group-theoretic invariants.

ABSTRACT

We give a complete classification of finitely generated virtually free groups up to $\forall\exists$-elementary equivalence. As a corollary, we give an algorithm that takes as input two finite presentations of virtually free groups, and decides whether these groups have the same $\forall\exists$-theory or not.

Motivation & Objective

• To classify finitely generated virtually free groups up to ∀∃-elementary equivalence.
• To determine necessary and sufficient conditions for two such groups to have the same ∀∃-theory.
• To develop a decision procedure for ∀∃-elementary equivalence using finite group presentations.
• To extend techniques from hyperbolic group theory to virtually free groups, particularly in the context of first-order logic and quantifier elimination.

Proposed method

• Use the characterization of virtually free groups as finite graphs of finite groups to analyze normalizers and centralizers of finite subgroups.
• Define 'legal large extensions' as HNN extensions over finite subgroups where the normalizer is non-elementary and the centralizer equals the subgroup.
• Define 'legal small extensions' as replacements of virtually cyclic subgroups by overgroups via K_G-nice embeddings.
• Construct multiple legal extensions of a group by iteratively applying legal large and small extensions to finite subgroups.
• Apply the main isomorphism algorithm from [DG11] to test isomorphism between constructed extensions of two input groups.
• Use the equivalence between ∀∃-elementary equivalence and mutual satisfaction of ∃∀-sentences of bounded complexity to verify theory equivalence algorithmically.

Experimental results

Research questions

• RQ1 When do two finitely generated virtually free groups have the same ∀∃-theory?
• RQ2 What structural transformations preserve ∀∃-elementary equivalence in virtually free groups?
• RQ3 Can ∀∃-elementary equivalence be algorithmically decided from finite presentations of virtually free groups?
• RQ4 How do torsion elements and normalizers of finite subgroups influence the ∀∃-theory of virtually free groups?
• RQ5 What is the role of HNN extensions and subgroup replacements in characterizing ∀∃-elementary equivalence?

Key findings

• Two finitely generated virtually free groups are ∀∃-elementarily equivalent if and only if they admit isomorphic multiple legal extensions via legal large and small extensions.
• A decidable algorithm exists to determine whether two finite presentations define groups with the same ∀∃-theory, relying on isomorphism testing of constructed extensions.
• The number of required legal extensions in the construction is bounded by a computable function of the group's finite subgroup conjugacy classes, normalizer ranks, and maximum finite subgroup order.
• The classification reveals that co-Hopfian virtually free groups are elementarily equivalent only if they are isomorphic, highlighting a sharp contrast with non-co-Hopfian cases.
• The results extend to hyperbolic groups, suggesting broader applicability in classifying hyperbolic groups up to elementary equivalence.
• The existence of ∃∀-elementary embeddings is characterized in terms of the same extension constructions, providing stronger logical embeddings than mere theory equivalence.