# Publications

# Research Papers

(1) Uniform Kazhdan constants and paradoxes of the plane (2019), *Transformation Groups (2020)*, ArXiv

## Abstract

Let $G=\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ and $H=\mathrm{SL}(2,\mathbb{Z})$. We prove that the action $G\curvearrowright\mathbb{R}^2$ is uniformly non-amenable and that the quasi-regular representation of $G$ on $\ell^2(G/H)$ has a uniform spectral gap. Both results are a consequence of a uniform quantitative form of ping-pong for affine transformations, which we establish here.(2) Arithmetic Groups and the Lehmer Conjecture (2020), with François Thilmany, accepted for publication, *Israel Journal of Mathematics* (2020), ArXiV

## Abstract

In this paper, we generalize a result of Sury and show that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups is equivalent to a weak form of Lehmer's conjecture. We also survey some related conjectures.# Book translations

(1) Introduction aux groupes arithmétiques (1969), by Armand Borel. University Lecture Series volume 73 (American Mathematical Society), AMS Bookstore