Arithmetic groups

A famous question of Lehmer (circa 1933, now known as the Lehmer conjecture) asks whether one can find irreducible integer polynomials which are not cyclotomic polynomials or trivial, and whose Mahler measure tends to 1. (The Mahler measure of a complex polynomial is the product of the norms of its roots which are > 1.)  This number-theoretical question has been studied by various mathematicians; progress has been made, and some restricted cases are known (e.g. for polynomials of bounded degree, or for palindromic polynomials). 

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In 1992, Sury discovered an important connection between the Lehmer conjecture restricted to polynomials with only one (real) root outside the unit circle, and the existence of a neighborhood of 1 in SL(2,R) which avoids every cocompact, torsion-free lattice. The same year, Neumann & Reid established the equivalence of this same restricted version of the Lehmer conjecture, and the short geodesic conjecture (which asserts that there is a uniform lower bound on the lengths of closed geodesics) for hyperbolic 2-manifolds. 

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In the talk, we will briefly survey some of these known results. Then we will discuss a generalization of Sury’s theorem (joint work with Lam Pham), which shows the equivalence between a weak version of the Lehmer conjecture and a conjecture of Margulis about the existence of a neighborhood of 1 avoiding torsion-free, cocompact lattices in arbitrary real Lie groups.