Unipotent translation-like actions on lattices in semisimple Lie groups

Abstract: The Gersten conjecture says that a group being hyperbolic is equivalent to having no Baumslag-Solitar subgroups. This is known to be false due to work of Brady. While there are some weaker versions still open, we are interested in a geometric reformulation of the Gersten conjecture using translation-like actions. To be more specific, the geometric Gersten conjecture asks whether hyperbolicity is equivalent to having no translation-like action by any Baumslag-solitar group. In this talk, we show that cocompact lattices in real semisimple Lie groups admit translation-like actions by cocompact lattices in the unipotent part of the Iwasawa decomposition of the original Lie group. In particular, we demonstrate that $Z_n$ acts translation-like on the fundamental group of any closed hyperbolic $(n+1)$-manifold which provides counterexamples to the geometric Gersten conjecture. This is joint work with Ben McReynolds.