Poincaré profiles of lamplighter diagonal products

Abstract: We exhibit finitely generated groups with prescribed Poincar\'e profiles. It can be prescribed for functions between $n/\log n$ and linear, and is sharp for functions at least $n/(\log\log n)$. Those profiles were introduced by Hume, Mackay and Tessera in 2019 as a generalization of the separation profile, defined by Benjamini, Schramm and Tim\'ar in 2012. The family of groups used is based on a construction of Brieussel and Zheng. As applications, we show that there exists bounded degrees graphs of asymptotic dimension one that do not coarsely embed in any finite product of bounded degrees trees, and exhibit hyperfinite sequences of graphs of arbitrary large distortion in $L^p$-spaces.