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Tuesday, March 18, 2014

**Abstract:** By Varopoulos' theorem, the fundamental group of a closed quasiregularly elliptic n-manifold (i.e. a manifold receiving a quasiregular map from the Euclidean n-space) has polynomial order of growth at most n. Thus, by Gromov's theorem, these groups are virtually nilpotent. The existence of the map can, however, be used to obtain further information on the fundamental group. I will discuss two recent results: (1) a result with Rami Luisto showing that maximal growth of the group implies the group to be virtually abelian; (2) a result with Enrico Le Donne showing that fundamental groups of closed BLD-elliptic manifolds are (in fact) always virtually abelian.