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Free submonoids of hyperbolic monoids

Matthias Hamann

Abstract

In this paper, we prove that infinite cancellative finitely generated hyperbolic monoids never contain $\mathbb N\times\mathbb N$ as a submonoid but that they contain an element of infinite order and, if they are elementary, then they also contain a free monoid of rank at least 2. As a corollary we obtain that the latter have exponential growth. We prove these results by analysing the monoid of self-embeddings of hyperbolic digraphs and proving fixed-point theorems for them.

Free submonoids of hyperbolic monoids

Abstract

In this paper, we prove that infinite cancellative finitely generated hyperbolic monoids never contain as a submonoid but that they contain an element of infinite order and, if they are elementary, then they also contain a free monoid of rank at least 2. As a corollary we obtain that the latter have exponential growth. We prove these results by analysing the monoid of self-embeddings of hyperbolic digraphs and proving fixed-point theorems for them.
Paper Structure (6 sections, 35 theorems, 67 equations)

This paper contains 6 sections, 35 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Hyperbolic digraphs
  3. Self-em-bed-ding s
  4. Hyperbolic monoids
  5. Growth of hyperbolic monoids
  6. Self-embeddings of graphs

Key Result

Proposition 2.1

H-HyperbolicDigraph Let $\delta\geq 0$ and let $D$ be a $\delta$-hyperbolic digraph that satisfies (itm_Bounded1) and (itm_Bounded2) for the function $\varphi\colon \mathbb{R}\to\mathbb{R}$.

Theorems & Definitions (65)

  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • ...and 55 more