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Boundaries of acylindrically hyperbolic groups and applications

Boundary amenability. Given a finitely generated group $$G$$, the Novikov conjecture for $$G$$ asks for topological invariance of higher order signatures of manifolds with fundamental group $$G$$. The two most important geometric tools towards the Novikov conjecture are the asymptotic dimension of a group and boundary amenability. Conjecturally, a group which admits a small acylindrical action on a separable hyperbolic graph is boundary amenable and thus satisfies the Novikov conjecture. A goal of this project is to make progress towards this conjecture using the invariant measures constructed by Maher-Tiozzo.

Isometric actions on $$l^p$$-spaces. Hyperbolic groups admit proper affine isometric actions on an $$l^p$$-space for some $$p>1$$, while for higher rank lattices any such action has a fixed point. Conjecturally, acylindrically hyperbolic groups should admit proper affine isometric actions on an $$l^p$$-space. A goal of the project is to settle this conjecture for interesting special classes of groups including the mapping class groups.

Rigidity of cross ratios. A cross ratio is a Hölder continuous function on the space of quadruples of pairwise distinct points in $$S^{n-1}$$that satisfies certain properties. Any negatively curved $$n$$-manifold $$M$$ defines a cross ratio on $$S^{n-1}$$. Conjecturally, $$\pi_1(M)$$ can have infinite index in the automorphism group of the cross ratio only when $$M$$ is locally symmetric. The goal is to attack this conjecture using dynamical properties of the action of $$\pi_1(M)$$.

## Team Members

Prof. Dr. Ursula Hamenstädt