11 research outputs found
Expansion in perfect groups
Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an
integer q, denote by Ga_q the subgroup of Ga consisting of the elements that
project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q
with respect to the generating set S form a family of expanders when q ranges
over square-free integers with large prime divisors if and only if the
connected component of the Zariski-closure of Ga is perfect.Comment: 62 pages, no figures, revision based on referee's comments: new ideas
are explained in more details in the introduction, typos corrected, results
and proofs unchange
Small doubling in groups
Let A be a subset of a group G = (G,.). We will survey the theory of sets A
with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}.
The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos
Centenary conferenc
Counting and effective rigidity in algebra and geometry
The purpose of this article is to produce effective versions of some rigidity
results in algebra and geometry. On the geometric side, we focus on the
spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic
hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum
determines the commensurability class of the 2-manifold (resp., 3-manifold). We
establish effective versions of these rigidity results by ensuring that, for
two incommensurable arithmetic manifolds of bounded volume, the length sets
(resp., the complex length sets) must disagree for a length that can be
explicitly bounded as a function of volume. We also prove an effective version
of a similar rigidity result established by the second author with Reid on a
surface analog of the length spectrum for hyperbolic 3-manifolds. These
effective results have corresponding algebraic analogs involving maximal
subfields and quaternion subalgebras of quaternion algebras. To prove these
effective rigidity results, we establish results on the asymptotic behavior of
certain algebraic and geometric counting functions which are of independent
interest.Comment: v.2, 39 pages. To appear in Invent. Mat
Local spectral gap in simple Lie groups and applications
We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action of Γ on G, whenever Γ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting works of Bourgain and Gamburd [BG06, BG10], and Benoist and de Saxcé [BdS14]. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique Γ-invariant finitely additive measure defined on all bounded measurable subsets of G