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

2025/05/27 by Ascari, Dario, Fruchter, Jonathan · 2 citations
#20F65 #20F67 #57M07 (Primary) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2505.20960

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'≤ 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.

Citations

• Virtual homology of limit groups and profinite rigidity of direct products
• Prescribed virtual homological torsion of 3-manifolds
• On the profinite rigidity of surface groups and surface words
• Virtual retraction properties in groups
• Relative cohomology theory for profinite groups
• Essential surfaces in graph pairs
• Virtual Homological Torsion of Closed Hyperbolic 3-manifolds
• Quasiconvexity and relatively hyperbolic groups that split
• Approximating L²-invariants and homology growth
• The asymptotic growth of torsion homology for arithmetic groups
• Immersing almost geodesic surfaces in a closed hyperbolic three manifold
• Hall's Theorem for limit groups

Cited by

• Quasi-convex surface subgroups in some one-relator groups with torsion
• Virtual homological torsion: abundance versus growth in books of I-bundles

Related

• Homological torsion growth in non-normal chains of graphs of free groups
• Essential surfaces in graph pairs
• From pure braid groups to hyperbolic groups
• Virtual retractions in free constructions
• Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups