Finiteness of homological filling functions. (English) Zbl 1494.20060

Fleming, Joshua W.; Martínez-Pedroza, Eduardo

Involve 11, No. 4, 569-583 (2018).

Summary: Let \(G\) be a group. For any \(\mathbb{Z}G\)-module \(M\) and any integer \(d>0\), we define a function \(\operatorname{FV}_{M}^{d+1}: \mathbb{N} \rightarrow \mathbb{N}\cup \{\infty\}\) 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 \(FP_{d+1}\) 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 \(FP_{d+1}\), our main result implies that its \((d+1)\)-dimensional homological filling function takes only finite values.

MSC:

20F65	Geometric group theory
20J05	Homological methods in group theory
57M07	Topological methods in group theory

Keywords:

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

References:

[1]	10.1073/pnas.1207377110 · Zbl 1300.57003
[2]	10.1515/jgth.1999.008 · Zbl 0927.20021
[3]	10.1007/978-3-662-12494-9
[4]	10.1007/978-1-4684-9327-6
[5]	10.1112/jlms/54.2.261 · Zbl 0861.20033
[6]	; Gromov, Geometric group theory, II. London Math. Soc. Lecture Note Ser., 182, 1 (1993)
[7]	10.1007/s11856-008-1070-6 · Zbl 1211.20038
[8]	10.4171/GGD/369 · Zbl 1388.20061
[9]	10.2140/agt.2010.10.1807 · Zbl 1202.20046
[10]	10.4153/CMB-2015-070-2 · Zbl 1360.20037
[11]	10.1515/jgth-2017-0020 · Zbl 1435.20059
[12]	10.1007/PL00004801 · Zbl 0954.20021
[13]	10.1090/memo/0843 · Zbl 1093.20025
[14]	10.4171/GGD/144 · Zbl 1263.20040
[15]	10.1007/s12220-015-9601-y · Zbl 1380.20045