# 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} }

. 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…

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## One Citation

### On the profinite rigidity of free and surface groups

- Mathematics
- 2022

. Let S be a free or surface group. We establish a Tits alternative for groups in the ﬁnite, soluble and p -genus of S . This generalises (and gives a new proof of) the analogous result of Baumslag…

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