Short incompressible graphs and -free groups
Florent Balacheff
Universitat Autònoma de Barcelona, Bellaterra, SpainWolfgang Pitsch
Universitat Autònoma de Barcelona, Bellaterra, Spain
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Abstract
Consider a finite connected -complex endowed with a piecewise Riemannian metric, and whose fundamental group is freely indecomposable, of rank at least , and in which every -generated subgroup is free. In this paper, we show that we can always find a connected graph such that (in short, a -incompressible graph) whose length satisfies the following curvature-free inequality: . This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence, we obtain that the volume entropy of such -complexes with unit area is always bounded away from zero.
Cite this article
Florent Balacheff, Wolfgang Pitsch, Short incompressible graphs and -free groups. Rev. Mat. Iberoam. (2024), published online first
DOI 10.4171/RMI/1477