Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs Articles uri icon

publication date

  • August 2018

issue

  • 5 (1750050)

volume

  • 20

international standard serial number (ISSN)

  • 0219-1997

electronic international standard serial number (EISSN)

  • 1793-6683

abstract

  • In this paper, we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary.

keywords

  • Gromov hyperbolicity; Cheeger isoperimetric constant; bounded local geometry