## Geometric invariants of groups and property R-infty

##### Abstract

In this thesis we study property R_\infty for some classes of finitely generated groups by the use of the BNS invariant \Sigma^1 and some other geometric tools. In the combinatorial chapters of the work (4, 5, 6, 10 and 11), we compute \Sigma^1 for the family of Generalized Solvable Baumslag-Solitar groups \Gamma_n and use it to obtain a new proof of R_\infty for them, by using Gonçalves and Kochloukova's paper. Then, we get nice information on finite index subgroups H of any \Gamma_n by finding suitable generators and a presentation, and by computing their \Sigma^1. This gives a new proof of R_\infty for H and for every finite direct product of such groups. We also show that no nilpotent quotients of the groups \Gamma_n have R_\infty. With a help of Cashen and Levitt's paper, we give an algorithmic classification of all possible shapes for \Sigma^1 of GBS and GBS_n groups and show how to use it to obtain some partial twisted-conjugacy information in some specific cases. Furthermore, we show that the existence of certain spherically convex and invariant k-dimensional polytopes in the character sphere of a finitely generated group G can guarantee R_\infty for G. In the geometric chapters (7 through 9), we study property R_\infty for hyperbolic and relatively hyperbolic groups. First, we give a didactic presentation of the (already known) proof of R_\infty for hyperbolic groups given by Levitt and Lustig (which also uses a paper from Paulin). Then, we expand and analyse the sketch of proof of R_\infty for relatively hyperbolic groups given by A. Fel'shtyn on his survey paper: we point out the valid arguments and difficulties of the proof, exhibit what would be a complete proof based on his sketch and show an example where the proof method doesn't work.

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