Coarse models of homogeneous spaces and translations-like actions

2019/11/27 by McReynolds, D. B., Pengitore, Mark
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1911.12175

Abstract

For finitely generated groups G and H equipped with word metrics, a translation-like action of H on G is a free action where each element of H moves elements of G a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group G that is not isogenous to SL(2,ℝ) admit translation-like actions by ℤ². This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical N of the Borel subgroup AN of G acts translation-like on any cocompact lattice in G. We also prove that for noncompact simple Lie groups G,H with H

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• Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions
• Translation equivalence in free groups