Speaker
Description
The volume entropy of a closed Riemannian manifold is a number which measures the exponential rate of growth of balls in the Riemannian universal cover of the manifold. Taking the infimum of the volume entropy over all Riemannian metrics (up to normalization), one gets a homotopy invariant of the manifold, the minimal volume entropy. This invariant behaves in a quite mysterious way: for example, we do not know if it is multiplicative under finite covers. However, in 1982 Gromov proved that it is an upper bound for the simplicial volume, another (better-behaved) homotopy invariant which, intuitively, measures the difficulty of representing the fundamental class of the manifold via singular simplices. Gromov himself raised the question whether the vanishing of the simplicial volume implies the vanishing of minimal volume entropy. We prove that for mapping tori over oriented closed connected 3-manifolds, which are known to have vanishing simplicial volume by a 2020 result of Bucher and Neofytidis, this is indeed the case. One of our main technique involves finding covers of small cardinality of the mapping tori composed by sets of polynomial growth. This is a joint work with Alberto Casali, Francesco Milizia and Marco Moraschini.
