Lam Pham

Lam Pham

Instructor in Mathematics, Brandeis University

Publications

Research Papers

Bottom of the Length Spectrum of Arithmetic Orbifolds (with Mikolaj Fraczyk), Transactions of the AMS (2023) (ArXiV)

Abstract We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from 1. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold $\Gamma\backslash X$ by showing the existence of a positive constant $\delta(X)>0$ such that squares of lengths of closed geodesics shorter than $\delta$ must be pairwise linearly dependent over $\mathbb{Q}$.

Arithmetic Groups and the Lehmer Conjecture, (with François Thilmany), Israel Journal of Mathematics (2021) (ArXiV)

Abstract We generalize a result of Sury [1992] and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis [1991]) is equivalent to a weak form of Lehmer’s conjecture. We include a short survey of related results and conjectures.

Uniform Kazhdan constants and paradoxes of the plane, Transformation Groups (2022)

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.

Logarithmic diameter bounds for some Cayley graphs (with Xin Zhang), Journal of Group Theory (2022)

Abstract Let $S\subset \mathrm{GL}(n,\mathbb{Z})$ be a finite symmetric set. We show that if the Zariski closure of $\Gamma=\langle S\rangle$ is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph $\mathrm{Cay}(\Gamma/\Gamma(q),\pi_{q}(S))$ is $O(\log q)$, where $q$ is an arbitrary positive integer, $\pi_{q}:\Gamma\to \Gamma/\Gamma(q)$ is the canonical projection induced by the reduction modulo $q$, and the implied constant depends only on $S$.

Book translations

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.