Deformations and rigidity of lattices in solvable Lie groups

TL;DR

This paper extends classical rigidity results to lattices in solvable Lie groups, showing under certain conditions their deformation spaces are finite and Hausdorff, with examples illustrating various possible complexities.

Contribution

It generalizes rigidity theorems for lattices in solvable Lie groups, establishing finiteness and Hausdorff properties of deformation spaces for Zariski-dense lattices.

Findings

Deformation space of Zariski-dense lattices is finite and Hausdorff when the maximal nilpotent normal subgroup is connected.

Every lattice virtually embeds as a Zariski-dense lattice with finite deformation space.

Counterexamples with countably infinite or uncountable deformation spaces are provided.

Abstract

Let $G$ be a simply connected, solvable Lie group and $\Gamma$ a lattice in $G$. The deformation space $\mathcal{D}(\Gamma,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(\Gamma,G)$ of all lattice embeddings of $\Gamma$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev and Sait\^o for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice $\Gamma$ in $G$ is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of $G$ is connected. This implies that every lattice in a solvable Lie group virtually embeds as a Zariski-dense lattice with finite deformation space. We give examples of solvable Lie groups $G$ which admit Zariski-dense lattices $\Gamma$ such that $\mathcal{D}(\Gamma,G)$ is countably infinite, and also examples where the maximal nilpotent normal subgroup of $G$ is connected and simultaneously $G$ has lattices with uncountable deformation space.