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

About From the American Mathematical Society: Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel's Introduction aux groupes arithmétiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces. The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments.