Minimal volume entropy and fiber growth

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> , and more generally of a finite simplicial complex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , vanishes or is positive. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> to lower dimensional simplicial complexes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> . This leads to a complete characterization of spaces with positive minimal volume entropy for finite simplicial complexes whose fundamental group has uniform exponential growth with no subgroup of intermediate growth. As pointed out to us by Vitali Kapovitch, these conditions are related to collapsing with Ricci curvature bounded below and lead to a refinement of Gromov’s isolation theorem. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.