• Corpus ID: 252596169

CHARACTERISING SURFACE GROUPS BY THEIR VIRTUAL SECOND BETTI NUMBER

@inproceedings{Fruchter2022CHARACTERISINGSG,
  title={CHARACTERISING SURFACE GROUPS BY THEIR VIRTUAL SECOND BETTI NUMBER},
  author={Jonathan Fruchter and Ismael Morales},
  year={2022}
}

• Jonathan Fruchter, Ismael Morales
• Published 2022
• Mathematics

. Define the virtual second betti number of a finitely generated group G as vb 2 ( G ) = sup { dim H 2 ( H ; Q ) | H ≤ G of finite index } ∈ Z ≥ 0 ∪ {∞} . We show that if G is a one-ended word-hyperbolic group obtained as the fundamental group of a graph of free groups with cyclic edge groups then vb 2 ( G ) is finite if and only if G is the fundamental group of a closed surface, in which case vb 2 ( G ) = 1 . We extend this result to limit groups and prove that the virtual second betti number of a limit group L is finite if and only if L is either free, free abelian or the fundamental group of a closed and connected surface. As an application we give an alternative proof to Wilton’s result which states that surface groups are determined among limit groups (and hyperbolic fundamental groups of graphs of free groups with cyclic edge groups) by their profinite completions. An appendix to this paper by Morales investigates relations between the virtual second betti number and pro-soluble groups, and explains why the results above imply that Jaikin-Zapirain’s methods for showing that groups in the pro-soluble genera of free and surface groups are residually- p cannot be used for other hyperbolic limit groups.