Groups and Geometry in the South East

18 June 2021
Piotr Przytycki, Elia Fioravanti, Rylee Lyman

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Tits Alternative in dimension 2


Piotr Przytycki (McGill)

A group G satisfies the Tits alternative if each of its finitely generated subgroups contains a non-abelian free group or is virtually solvable. I will sketch a proof of a theorem saying that if G acts geometrically on a simply connected nonpositively curved complex built of equilateral triangles, then it satisfies the Tits alternative. This is joint work with Damian Osajda.

Coarse-median preserving automorphisms


Elia Fioravanti (Bonn)

We study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If Phi is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix(Phi) is finitely generated and undistorted. Up to replacing Phi with a power, the fixed subgroup is actually quasi-convex with respect to the standard word metric (which implies that it is separable and a virtual retract, by work of Haglund and Wise). Our techniques also apply to automorphisms of hyperbolic groups and to certain automorphisms of hierarchically hyperbolic groups. Based on arXiv:2101.04415.

Some new CAT(0) free-by-cyclic groups


Rylee Lyman (Rutgers-Newark)

I will construct several infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten's example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.


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