On cusp holonomies in strictly convex projective geometry

TL;DR

This paper characterizes cusp holonomies in convex projective geometry, constructs generalized cusps with various fundamental groups, and explores implications for Anosov representations.

Contribution

It provides a complete classification of cusp holonomies, extends the notion of generalized cusps to include virtually solvable groups, and constructs examples with non-virtually nilpotent fundamental groups.

Findings

Complete characterization of cusp holonomies.

Construction of generalized cusps with diverse fundamental groups.

First example of a generalized cusp with non-virtually nilpotent fundamental group.

Abstract

We give a complete characterization of the holonomies of strictly convex cusps and of round cusps in convex projective geometry. We build families of generalized cusps of non-maximal rank associated to each strictly convex or round cusp. We also extend Ballas-Cooper-Leitner's definition of generalized cusp to allow for virtually solvable fundamental group, and we produce the first such example with non-virtually nilpotent fundamental group. Along with a companion paper, this allows to build strictly convex cusps and generalized cusps whose fundamental group is any finitely generated virtually nilpotent group. This also has interesting consequences for the theory of relatively Anosov representations.