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
    Project leader
    Rheinische Friedrich-Wilhelms-Universität Bonn

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