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We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of… (More)
This is an informal announcement of results to be described and proved in detail in . We give various results on the structure of approximate subgroups in linear groups such as SLn(k). For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SLn(Fq) which generates the group must be… (More)
We show that any locally compact group G with polynomial growth is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We then study the shape of large balls and show, generalizing work of P. Pansu, that after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. As a… (More)
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group.… (More)
We show that for every integer d ∈ N, there is N(d) ∈ N such that if K is any field and F is a finite subset of GLd(K), which generates a non amenable subgroup, then F contains two elements, which freely generate a non abelian free subgroup. This improves the original statement of the Tits alternative. It also implies a growth gap and a co-growth gap for… (More)
We discuss free subsemigroups in solvable and elementary amenable groups improving earlier results of Milnor-Wolf, Rosenblatt, Alperin and Osin about uniform exponential growth for these groups.
We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height ĥ(F ) of a finite set of matrices F in SLn(Q) which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois… (More)
We describe the structure of “K-approximate subgroups” of solvable subgroups of GLn(C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsion-free nilpotent groups , we show that such approximate subgroups are efficiently controlled by nilpotent progressions.
There are many excellent existing texts for the material in this lecture, starting with Lubotzky's monograph  and recent AMS survey paper . For expander graphs and their use in theoretical computer science, check the survey by Hoory, Linial and Wigderson . We give here a brief introduction. We start with a definition. Definition 0.1. (Expander… (More)