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Short incompressible graphs and $2$-free groups

Florent Balacheff, Wolfgang Pitsch

Abstract

Consider a finite connected $2$-complex $X$ endowed with a piecewise Riemannian metric and whose fundamental group is freely indecomposable, of rank at least $3$, and in which every $2$-generated subgroup is free. In this paper we show that we can always find a connected graph $Γ\subset X$ such that $π_1 Γ\simeq {\mathbb F}_2 \hookrightarrowπ_1 X$ (in short, a $2$-incompressible graph) whose length satisfies the following curvature-free inequality: $\ell(Γ)\leq 4\sqrt{2\text{Area}(X)}$. This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence we obtain that the volume entropy of such $2$-complexes with unit area is always bounded away from zero.

Short incompressible graphs and $2$-free groups

Abstract

Consider a finite connected -complex endowed with a piecewise Riemannian metric and whose fundamental group is freely indecomposable, of rank at least , and in which every -generated subgroup is free. In this paper we show that we can always find a connected graph such that (in short, a -incompressible graph) whose length satisfies the following curvature-free inequality: . This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence we obtain that the volume entropy of such -complexes with unit area is always bounded away from zero.
Paper Structure (4 sections, 8 theorems, 27 equations)

This paper contains 4 sections, 8 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Topology of small balls in piecewise flat $2$-complexes
  3. Geometry of small balls in piecewise flat $2$-complexes
  4. A Universal bound for the volume entropy

Key Result

Theorem 1.1

Any $2$-free group $G$ which is freely indecomposable and of rank at least $3$ satisfies the following inequality:

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • ...and 4 more