Margulis and Soifer proved that every finitely generated linear group which is not virtually solvable has uncountably many infinite index maximal subgroups. However, not much is known about the structure of these subgroups. We will construct infinite index maximal subgroups of $SL(n , \mathbb{Z})$ whose actions on projective space have different dynamical properties and discuss some algebraic consequences of these constructions. This is joint work with Tsachik Gelander.
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