Homogeneous number of free generators
13/12/2013 Sextafeira, 13 de Dezembro de 2013, 14h4515h45, IIIULSala B101
Gregory Soifer (Faculty of Exact Sciences, BarIlan University)
Institute for Interdisciplinary Research  University of Lisbon
First we will recall ideas, conjectures and results on free subgroups in linear groups. Then I will talk about our resent result with Menny Aka and Tsachik Gelander which gives an answer on two questions of Simon Thomas. Namely, we show that for any $n\geq 3$ one can find a four generated free subgroup of $SL_{n}(\mathbb{Z})$ which is profinitely dense. More generally, we show that an arithmetic group $\Gamma$ which admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank.
Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group $\Gamma$ is uncountable.
