Quasisymmetric maps of boundaries of amenable hyperbolic groups

2014/04/21 by Dymarz, Tullia
#FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1404.5099

Abstract

In this paper we show that if Y=N × ℚ_m is a metric space where N is a Carnot group endowed with the Carnot-Caratheodory metric then any quasisymmetric map of Y is actually bilipschitz. The key observation is that Y is the parabolic visual boundary of a mixed type locally compact amenable hyperbolic group. The same results also hold for a larger class of nilpotent Lie groups N. As part of the proof we also obtain partial quasi-isometric rigidity results for mixed type locally compact amenable hyperbolic groups. Finally we prove a rigidity result for uniform subgroups of bilipschitz maps of Y in the case of N= ℝⁿ.

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