Constructing proper affine actions via higher strip deformations

TL;DR

This paper introduces higher strip deformations to construct proper affine actions of free groups in higher dimensions, expanding the class of groups known to act properly on affine spaces.

Contribution

It develops a new method using higher strip deformations and the Margulis invariant to produce proper affine actions of free and virtually free groups in higher-dimensional spaces.

Findings

Proper affine actions of convex cocompact free groups in d44n-1 dimensions.

Construction of proper actions of the group C2 * C3 on d47.

Introduction of higher strip deformations as a novel construction technique.

Abstract

We introduce higher strip deformations, which give a way of constructing affine deformations of discrete free groups in the image of the irreducible representation $\operatorname{PSL}_2\mathbb{R}\to \operatorname{SO}(2n,2n-1)$. We use the Margulis invariant to find a properness condition for these deformations, showing that each convex cocompact free group admits affine deformations acting properly on $\mathbb{R}^{4n-1}$. Using our method we can also construct proper affine actions of virtually free groups, which we demonstrate by finding proper actions of $C_2 \star C_3$ on $\mathbb{R}^7.$