Finiteness of homological filling functions

Fleming, Joshua W.; Martínez-Pedroza, Eduardo

doi:10.2140/involve.2018.11.569

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Finiteness of homological filling functions

Joshua W. Fleming and Eduardo Martínez-Pedroza

Vol. 11 (2018), No. 4, 569–583

DOI: 10.2140/involve.2018.11.569

Abstract

Let $G$ be a group. For any $ℤ G$ -module $M$ and any integer $d > 0$ , we define a function ${FV}_{M}^{d + 1} : ℕ \to ℕ \cup {\infty}$ generalizing the notion of $(d + 1)$ -dimensional filling function of a group. We prove that this function takes only finite values if $M$ is of type $F P_{d + 1}$ and $d > 0$ , and remark that the asymptotic growth class of this function is an invariant of $M$ . In the particular case that $G$ is a group of type $F P_{d + 1}$ , our main result implies that its $(d + 1)$ -dimensional homological filling function takes only finite values.

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Keywords

Dehn functions, homological filling function, isoperimetric inequalities, finiteness properties of groups

Mathematical Subject Classification 2010

Primary: 20F65, 20J05

Secondary: 16P99, 28A75, 57M07

Milestones

Received: 15 September 2016

Accepted: 22 July 2017

Published: 15 January 2018

Communicated by Kenneth S. Berenhaut

Authors

Joshua W. Fleming
Memorial University of Newfoundland St. John’s, NL Canada

Eduardo Martínez-Pedroza
Memorial University of Newfoundland St. John’s, NL Canada

Vol. 11, No. 4, 2018