Combinatorial higher dimensional isoperimetry and divergence

2015/07/06 by Jason Behrstock, Behrstock, Jason, Cornelia Drutu +1
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Metric Geometry (math.MG) #math.GR #math.GT #math.MG

paper · pdf · doi:10.48550/arxiv.1507.01518

This paper contains material that formerly formed the first half of arXiv:1305.2994, as well as strengthening and refinements of those results

arxiv created 2015/07/06 · arxiv updated 2015/07/07

Abstract

In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called "round" and "unfolded", provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer--Fleming inequality for finitely generated groups, the construction of examples of CAT(0)--groups with higher dimensional divergence equivalent to x^d for every degree d [arXiv:1305.2994], and a proof of the fact that for bi-combable groups the filling function above the quasi-flat rank is asymptotically linear [Behrstock-Drutu].

Citations

• Higher dimensional divergence for mapping class groups

Related

• A bicombing that implies a sub-exponential isoperimetric inequality
• On the Isoperimetric Inequality in Finitely Generated Groups
• Combinatorial isoperimetric inequality for the free factor complex
• Isoperimetry in Finitely Generated Groups
• Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups