### The (non-)hyperbolic geometry of Gromov's monsters

Analysis and Geometry Seminar

5th June 2018, 3:00 pm – 4:00 pm

Howard House, 2nd Floor Seminar Room

Gromov's monsters are finitely generated groups whose Cayley graphs contain expander graphs in a reasonable geometric sense - a property of great interest in geometric and analytic group theory. There are two known types of these groups - one type arises from Gromov's graphical model for random groups, the other as a class of graphical small cancellation groups. In this talk, I will discuss recent results showing that - despite many similarities in their constructions - the two types of Gromov's monsters are in fact very different. I will explain that graphical small cancellation groups are acylindrically hyperbolic and thus have many properties of hyperbolic groups, whereas Gromov's random groups are quite the opposite: they cannot act non-elementarily on any Gromov hyperbolic space. Furthermore, the divergence functions of Gromov's random groups are linear on a subsequence - thus they cannot even be quasi-isometric to any acylindrically hyperbolic group. All concepts will be explained in the talk.

Based on joint works with Alessandro Sisto and Romain Tessera.

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