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Homological Isoperimetric Inequalities for Kernels of Free Extensions of Type $FP_2$

Jakub F. Tucker

Abstract

We define homological area-radius pairs with surface diagrams. Using these, we adapt a proof of Gersten and Short \cite{gersten2002} to obtain a homological isoperimetric inequality for subgroups of type $FP_2$ which appear as kernels of free extensions.

Homological Isoperimetric Inequalities for Kernels of Free Extensions of Type $FP_2$

Abstract

We define homological area-radius pairs with surface diagrams. Using these, we adapt a proof of Gersten and Short \cite{gersten2002} to obtain a homological isoperimetric inequality for subgroups of type which appear as kernels of free extensions.

Paper Structure

This paper contains 11 sections, 15 theorems, 15 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Homological Dehn Function
  3. Finitely Presented Kernels of Free Extensions
  4. Kernels of type FP2
  5. Surface Diagrams
  6. Homological Area-Radius Pairs
  7. A Homological Finite Presentation for H
  8. New Filling from an Automorphism
  9. t-rings on Surface Diagrams
  10. The Homological Isoperimetric Inequality
  11. Hyperbolic Groups

Key Result

Theorem 1.1

Given an extension where $F_n$ is free and $K$ is finitely presented, then if $H$ is hyperbolic, $K$ satisfies a polynomial isoperimetric inequality. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Gluing edges around $h$, highlighted edges may be glued with the orientation shown.
  • Figure 2: Near $h$ is a surface (with boundary).
  • Figure 3: Obtaining the surface diagram $S'$ by refining a surface diagram $S$.
  • Figure 4: An internal $A$-edge.
  • Figure 5: Sketch of part of $\Gamma(H,A\cup\left\{t\right\})$, where $n=1$.
  • ...and 1 more figures

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 2.1: brady2021
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2.2: brady2021
  • Proposition 2.3: brady2021
  • Definition 2.4: brady2021
  • Definition 2.5
  • Remark 4
  • ...and 31 more