Szabo Endre: Növekedés Lie típusú csoportokban.

We prove that if $L$ is a finite simple group of Lie type and $A$ a symmetric set of generators of $L$, then $A$ grows i.e $|AAA| > |A|^{1+\e}$ where $\e$ depends only on the Lie rank of $L$, or $AAA=L$. This implies that for a family of simple groups $L$ of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in $|L|$.

Combining our result on growth with known results of Bourgain, Gamburd and Varj\'u it follows that if $\Lambda$ is a Zariski-dense subgroup of $SL(d,\BZ)$ generated by a finite symmetric set $S$, then for square-free moduli $m$, which are relatively prime to some number $m_0$, the Cayley graphs $\Gamma(SL(d,\BZ/m\BZ),\pi_m(S))$ form an expander family.