# A Strong Tits Alternative

@article{Breuillard2008AST, title={A Strong Tits Alternative}, author={Emmanuel Breuillard}, journal={arXiv: Group Theory}, year={2008} }

We show that for every integer $d$, there is a constant $N(d)$ such that if $K$ is any field and $F$ is a finite subset of $GL_d(K)$, which generates a non amenable subgroup, then $F^{N(d)}$ 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 non-amenable linear groups, and has consequences about the girth and uniform expansion of small sets in finite…

## 37 Citations

A Height Gap Theorem For Finite Subsets Of GL_d(\bar{Q}) and Non Amenable Subgroups

- Mathematics
- 2008

We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\hat{h}(F)$ of a finite set of matrices $F$ in…

Approximate subgroups of residually nilpotent groups

- MathematicsMathematische annalen
- 2019

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step.…

Approximate Subgroups of Linear Groups

- Mathematics
- 2010

We establish various results on the structure of approximate subgroups in linear groups such as SLn(k) that were previously announced by the authors. For example, generalising a result of Helfgott…

N ov 2 01 1 A HEIGHT GAP THEOREM FOR FINITE SUBSETS OF GL d ( Q ) AND NON AMENABLE SUBGROUPS

- Mathematics
- 2021

We introduce a conjugation invariant normalized height ĥ(F ) on finite subsets of matrices F in GLd(Q) and describe its properties. In particular, we prove an analogue of the Lehmer problem for this…

Uniform exponential growth for CAT(0) square complexes

- MathematicsAlgebraic & Geometric Topology
- 2019

In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove…

Expansion in SL 2$${(\mathbb{R})}$$ and monotone expanders

- Mathematics
- 2013

This work presents an explicit construction of a family of monotone expanders, which are bi-partite expander graphs whose edge-set is defined by (partial) monotone functions. The family is (roughly)…

Expansion, random walks and sieving in $$S{L_2}({\mathbb{F}_p}[t])$$

- Mathematics
- 2015

We construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of $$S{L_2}({\mathbb{F}_p}[t])$$ modulo certain square-free ideals.…

Monotone expansion

- MathematicsSTOC '12
- 2012

This work presents an explicit construction of a family of monotone expanders, which are bi-partite expander graphs whose edge-set is defined by (partial)monotone functions, and shows a product-growth theorem for SL2(R); roughly, that for every A ⊂ SL2 (R) with certain properties, the size of AAA is much larger than that of A.

Ping-pong in Hadamard manifolds

- Mathematics
- 2019

Author(s): Dey, S; Kapovich, M; Liu, B | Abstract: In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a…

Local spectral gap in simple Lie groups and applications

- Mathematics
- 2015

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 $$\Gamma \curvearrowright…

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