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Homological Filling Functions with Coefficients

Xingzhe Li, Fedor Manin

Abstract

How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in "Asymptotic invariants of infinite groups", we define homological filling functions of groups with coefficients in a group $R$. Our main theorem is that the coefficients make a difference. That is, for every $n \geq 1$ and every pair of coefficient groups $A, B \in \{\mathbb{Z},\mathbb{Q}\} \cup \{\mathbb{Z}/p\mathbb{Z} : p\text{ prime}\}$, there is a group whose filling functions for $n$-cycles with coefficients in $A$ and $B$ have different asymptotic behavior.

Homological Filling Functions with Coefficients

Abstract

How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in "Asymptotic invariants of infinite groups", we define homological filling functions of groups with coefficients in a group . Our main theorem is that the coefficients make a difference. That is, for every and every pair of coefficient groups , there is a group whose filling functions for -cycles with coefficients in and have different asymptotic behavior.

Paper Structure

This paper contains 15 sections, 19 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Why filling functions with coefficients?
  3. Proof methods
  4. Preliminaries
  5. Definition of homological filling functions with coefficients
  6. Right-angled Artin groups
  7. Two general lemmas
  8. The case $n=1$
  9. Constructing the groups $G$ and $H$
  10. Properties of $G$ and $H$
  11. Upper bound on homological Dehn functions
  12. Lower bound on homological Dehn functions
  13. Higher dimensions
  14. Construction and properties
  15. Bounds on homological filling functions

Key Result

Theorem A

Let $q$ be a prime, $n \geq 1$, and $d \in \mathbb{N} \cup \{\infty\}$. Then there is a group $H$ of type $\mathcal{F}^{n+1}$ (a normal subgroup of an $(n+2)$-dimensional CAT(0) group) such that where $f_{d,n}(x)=\exp(\!\sqrt[n]{x})$ if $d=\infty$ and $x^{d/n}$ otherwise.

Figures (1)

  • Figure 1: A schematic illustration of the space $\widetilde{Z}$ and a typical cycle inside it. We illustrate the process of reflecting a piece of the cycle across a component of $\widetilde{W}$.

Theorems & Definitions (35)

  • Theorem A
  • Theorem B
  • Proposition 2.1
  • proof
  • Theorem 2.2: Bestvina and Brady BB
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • ...and 25 more