2006/08/31 by Noel Brady, Martin R. Bridson, Martin Bridson +2
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics #Eigenvalues and eigenvectors #Function (biology) #Geometric and Algebraic Topology #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Integer (computer science) #Isoperimetric inequality #Mathematics #Pure mathematics #math.GR #math.GT #msc:20E06 #msc:20F65 #msc:20F69 #msc:53C99 #msc:57M07 #msc:57M20
paper · pdf · doi:10.2140/gt.2009.13.141
published as Geom. Topol. 13 (2009) 141-187 · 42 pages, 8 figures. Version 2: 47 pages, 8 figures; minor revisions and reformatting; to appear in Geom. Topol
arxiv created 2008/09/17 · openalex publication_date 2009/01/01 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The k -dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k -spheres mapped into k -connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each nonnegative integer matrix P and positive rational number r , we associate a finite, aspherical 2-complex X r;P and determine the Dehn function of its fundamental group G r;P in terms of r and the Perron-Frobenius eigenvalue of P . The range of functions obtained includes .x/ D x s , where s 2 Q OE2; 1/ is arbitrary. Next, special features of the groups G r;P allow us to construct iterated multiple HNN extensions which exhibit similar isoperimetric behavior in higher dimensions. In particular, for each positive integer k and rational s > .k C 1/=k , there exists a group with k -dimensional Dehn function x s . Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs .M; @M / in addition to .B kC1 ; S k /.