The joint translation spectrum and Manhattan manifolds

TL;DR

This paper introduces geometric versions of the Benoist limit cone and joint spectrum, called the translation cone and joint translation spectrum, linking them to Manhattan manifolds and random walk drift vectors.

Contribution

It defines new geometric notions generalizing existing concepts and connects the joint translation spectrum to the gradient of Manhattan manifolds, advancing the understanding of group actions and spectra.

Findings

The joint translation spectrum equals the image of the gradient function of a Manhattan manifold.

The spectrum is the closure of drift vectors from finitely supported symmetric random walks.

Properties of the spectrum are deduced from its geometric and probabilistic interpretations.

Abstract

We define and study geometric versions of the Benoist limit cone and matrix joint spectrum, which we call the translation cone and the joint translation spectrum, respectively. These new notions allow us to generalize the study of embeddings into products of rank-one simple Lie groups and to compare group actions on different metric spaces, quasi-morphisms, Anosov representations and many other natural objects of study. We identify the joint translation spectrum with the image of the gradient function of a corresponding Manhattan manifold: a higher dimensional version of the well known and studied Manhattan curve. As a consequence we deduce many properties of the spectrum. For example we show that it is given by the closure of the set of all possible drift vectors associated to finitely supported, symmetric, admissible random walks on the associated group.