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Virtual homological torsion in graphs of free groups with cyclic edge groups

Dario Ascari, Jonathan Fruchter

TL;DR

The paper studies virtual torsion in the abelianization of finite-index subgroups of hyperbolic groups that split as graphs of free groups with cyclic edge groups, proving that any finite abelian group $M$ appears as a direct summand unless the group is a free product of free and surface groups. The authors develop a framework based on branched surfaces and systems of $\\\partial$-equations, together with precovers and JSJ-type decompositions, to realize prescribed torsion in finite covers. This leads to applications in profinite rigidity, showing that free products of free and surface groups are profinitely rigid within the class, and to rigidity statements for partial surface words via word measures on finite groups. The methods bridge combinatorial-topological constructions with number-theoretic invariants, providing a robust approach to virtual torsion phenomena in a broad class of hyperbolic groups and connecting to broader themes in 3-manifold topology and profinite group theory.

Abstract

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in the abelianization of some finite-index subgroup $G'\le G$. As an application, we deduce that free products of free and surface groups are profinitely rigid among hyperbolic graphs of free groups with cyclic edge groups. We also conclude that partial surface words in a free group are determined by the word measures they induce on finite groups.

Virtual homological torsion in graphs of free groups with cyclic edge groups

TL;DR

The paper studies virtual torsion in the abelianization of finite-index subgroups of hyperbolic groups that split as graphs of free groups with cyclic edge groups, proving that any finite abelian group appears as a direct summand unless the group is a free product of free and surface groups. The authors develop a framework based on branched surfaces and systems of -equations, together with precovers and JSJ-type decompositions, to realize prescribed torsion in finite covers. This leads to applications in profinite rigidity, showing that free products of free and surface groups are profinitely rigid within the class, and to rigidity statements for partial surface words via word measures on finite groups. The methods bridge combinatorial-topological constructions with number-theoretic invariants, providing a robust approach to virtual torsion phenomena in a broad class of hyperbolic groups and connecting to broader themes in 3-manifold topology and profinite group theory.

Abstract

Let be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless is isomorphic to a free product of free and surface groups, every finite abelian group appears as a direct summand in the abelianization of some finite-index subgroup . As an application, we deduce that free products of free and surface groups are profinitely rigid among hyperbolic graphs of free groups with cyclic edge groups. We also conclude that partial surface words in a free group are determined by the word measures they induce on finite groups.

Paper Structure

This paper contains 21 sections, 35 theorems, 51 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs of groups (and spaces)
  4. Covers, precovers and (geometric) elevations
  5. Separability properties
  6. Free groups and pairs
  7. Pairs and peripheral structures on free groups
  8. Peripheral structures on surfaces with boundary
  9. Pair splittings and JSJ decompositions
  10. Clean covers and free splittings
  11. Normal form
  12. Surface-type subpairs
  13. ∂-equations and branched surfaces
  14. Branched surfaces
  15. Standard branched surfaces
  16. ...and 6 more sections

Key Result

Theorem 1

Let $G$ be a hyperbolic group that splits as a graph of free groups amalgamated along cyclic subgroups, and that is not isomorphic to a free product of free and surface groups. Then for every finite abelian group $M$, there exists a finite-index subgroup $H\le G$ such that $M$ is a direct summand of

Figures (6)

  • Figure 1: Let $X$ be the rose graph with two petal and let $F=\langle a,b \rangle$ be its fundamental group. Let $w_1=a$ and $w_2=[a,b]$, and let $\underline{w}:S^1 \sqcup S^1 \rightarrow X$ be the map realizing the conjugacy classes $[w_1]$ and $[w_2]$. The (pullback) diagram above describes the elevations of $[\underline{w}]$ to the subgroup $H=\langle a^3, [a,b],ba^2b^{-1} \rangle$ of $F$. The graph $Y$ at the top-right corner, whose basepoint appears in green, is the core graph of the cover corresponding $H$. The three elevations corresponding to the conjugacy classes $[a^3]$ and $[ba^2b^{-1}]_H$ (of degrees $3$ and $2$ over $[a]_F$ respectively) and $[[a,b]]_H$ (degree $1$ over $[[a,b]]_F$) form the map $\underline{u}:S^1 \sqcup S^1 \sqcup S^1 \rightarrow Y$.
  • Figure 2: A branched surface $\mathcal{B}$. The branching locus $\mathcal{C}$ is highlighted in green and in red (the green connected component is of branching valence $4$, and the red connected component is of branching valence $3$). Note that the blue curve (where two surface pieces are glued) is not a part of the branching locus.
  • Figure 3: A standard branched surface.
  • Figure 4: The cover $\mathcal{B}_{\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}}$ of $\mathcal{B}$. The standard branched surface $\mathcal{B}$ appears on the right, and its $11$-sheeted cover $\mathcal{B}_{\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}}$ constructed in \ref{['prop:branched-torsion']} appears on the left.
  • Figure 5: An example of the procedure described in \ref{['prop:surfaces-prescribed-peripherals']}. Rigid pieces are represented as rectangles, and the surfaces inside the rectangles are the ones obtained using Wilton's \ref{['thm:Wilton-surfaces']}. Cyclic vertices appear in bold blue. The surface $\mathcal{H}$, obtained by gluing several of these pieces, appears in bold, and has two boundary components (hanging elevations) $[t_1]$ and $[t_2]$.
  • ...and 1 more figures

Theorems & Definitions (103)

  • Theorem 1: \ref{['thm:main']}
  • Corollary 2: \ref{['cor:virtual']}
  • Theorem 3: \ref{['thm:pf-rigid']}
  • Conjecture
  • Corollary 4: \ref{['cor:partial_words']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 93 more