** "Geometry and Analysis on Groups" Research Seminar **

**Title:**
Poincaré profile of connected Lie groups.

**Speaker:**
Romain Tessera (Université Paris-Sud)

**Abstract:**
We introduce a new family of geometric invariants for infinite graphs, and more generally metric measure spaces. Its main strength is that it is monotonous under coarse embedding. For every
\(n\)
, the
\(L^p\)
-Poincaré profile is
\(n\)
times the maximum Cheeger constant of a subgraph of size
\(n\)
. We compute it for all unimodular connected Lie groups, showing that it is either polynomial or grows like
\(n/\log n\). The computation is especially interesting for rank 1 simple Lie groups. We deduce various new non-embeddability results.