A simultaneous Abels-Margulis-Soifer lemma

Abstract

The Abels-Margulis-Soifer lemma states that if a semigroup $\Gamma$ acts strongly irreducibly by linear transformations on a finite-dimensional real vector space, then any element of $\Gamma$ can be multiplied by an element of some fixed finite subset of $\Gamma$ so that it becomes proximal (i.e. it acts on the corresponding projective space with an attracting fixed point and a repelling projective hyperplane) and even uniformly proximal (i.e. the distance between the attracting fixed point and the repelling projective hyperplane is uniformly bounded from below and the contraction towards the attracting fixed point is uniformly strong). We prove a version of this lemma simultaneously for linear representations of a semigroup $\Gamma$, acting on the corresponding projective spaces, and for representations of $\Gamma$ to isometry groups of (not necessarily proper) Gromov hyperbolic metric spaces, acting on the corresponding Gromov boundaries.