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Author: Jonas Beyrer6
A collar lemma for partially hyperconvex surface group representations

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Authors: Jonas Beyrer, Beatrice Pozzetti

Date Published: 15th Jul 2021

Publication Type: Journal

Cross ratios on CAT(0) cube complexes and marked length-spectrum rigidity

Abstract (Expand)

We show that group actions on irreducible \CAT cube complexes with no free faces are uniquely determined by their \ell^1 length function. Actions are allowed to be non-proper and non-cocompact, as long … as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of \emphnon-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces). As our main tool, we develop a notion of cross ratio on Roller boundaries of \CAT cube complexes. Inspired by results in \emphnegative curvature, we give a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. The core of our work is then to show that, when there are no free faces, these cross-ratio preserving maps always extend to cubical isomorphisms. All our results equally apply to cube complexes with variable edge lengths. As a special case of our work, we construct a compactification of the Char\-ney–Stambaugh–Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler–Vogtmann Outer Space.

Authors: Jonas Beyrer, Elia Fioravanti

Date Published: 2019

Publication Type: Misc

Marked length spectrum rigidity for actions on CAT(0) cube complexes

Abstract (Expand)

Given an action on a metric space one can associate to each element of the group its translation length. This gives a function of the group to the reals called the marked length spectrum. Adding …ked length spectrum. Adding requirements for space and action, it is a natural question if the marked length spectrum already uniquely defines space and action. In this talk we want to show that this is the case when considering CAT(0) cube complexes (under some natural assumptions). The main tool to prove this will be a boundary rigidity using cross ratios. Joint workwith Elia Fioravanti.

Author: Jonas Beyrer

Date Published: 2019

Publication Type: Journal

A complete description of the antipodal set of most symmetric spaces of compact type

Abstract (Expand)

It is known that the antipodal set of a Riemannian symmetric space of compact type G/K consists of a union of K-orbits. We determine the dimensions of these K-orbits of most irreducible symmetric spaces … of compact type. The symmetric spaces we are not going to deal with are those with restricted root system \mathfrak a_r and a non-trivial fundamental group, which is not isomorphic to \mathbb Z_2 or \mathbb Z_r+1. For example, we show that the antipodal sets of the Lie groups Spin(2r+1)r≥5, E_8 and G_2 consist only of one orbit which is of dimension 2r, 128 and 6, respectively; \SO(2r+1) has also an antipodal set of dimension 2r; and the Grassmannian Gr_r,r+q(R) has a rq-dimensional orbit as antipodal set if r≥5 and r≠q>0.

Author: Jonas Beyrer

Date Published: 2018

Publication Type: Journal

Cubulations and cross ratios on contracting boundaries

Abstract (Expand)

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations … and geodesic currents. We add to this picture by studying cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary \partial_∞G. A consequence of our results is that essential, hyperplane-essential cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work in arXiv:1903.02447. Along the way, we describe the relationship between the Roller boundary of a \mathrmCAT(0) cube complex, its Gromov boundary and - in the non-hyperbolic case - the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Authors: Jonas Beyrer, Elia Fioravanti

Date Published: 2018

Publication Type: Journal

CAT(0) cube complexes are determined by their boundary cross ratio

Abstract (Expand)

We introduce a \Z–valued cross ratio on Roller boundaries of \mathrmCAT cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely … extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group.

Authors: Jonas Beyrer, Elia Fioravanti, Merlin Incerti-Medici

Date Published: 2018

Publication Type: Misc

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