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Groups of type $FP$ via graphical small cancellation

Thomas Brown, Ian J Leary

Abstract

We construct an uncountable family of groups of type $FP$. In contrast to every previous construction of non-finitely presented groups of type $FP$ we do not use Morse theory on cubical complexes; instead we use Gromov's graphical small cancellation theory.

Groups of type $FP$ via graphical small cancellation

Abstract

We construct an uncountable family of groups of type . In contrast to every previous construction of non-finitely presented groups of type we do not use Morse theory on cubical complexes; instead we use Gromov's graphical small cancellation theory.

Paper Structure

This paper contains 8 sections, 23 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Using graphical small cancellation
  4. Graphs of groups
  5. Embedding polygon subgroups
  6. The projective line and its symmetries
  7. A spectacular 2-complex
  8. Closing remarks

Key Result

Theorem 1.2

For each $S\subseteq Z$ the graphical presentation for $H(S)$ given above satisfies the graphical small cancellation condition $C'(1/6)$. For each $S\subsetneq T\subseteq Z$ the natural bijection between generating sets extends to a surjective group homomorphism $H(S)\rightarrow H(T)$, whose kernel

Figures (2)

  • Figure 1: A graphical relator and its degree $2$ and $-2$ subdivisions.
  • Figure 2: A star-shaped graph of groups and its Eilenberg--Mac Lane space.

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • ...and 40 more