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Trees, homology, and automorphism groups of RAAGs

Javier Aramayona, José L. Fernández, Pablo Fernández, Conchita Martínez-Pérez

Abstract

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this subgroup, based on the number and degree of a certain type of vertices, which we call deep. We then use combinatorial methods to analyze the average value of this Betti number, in terms of the size of the defining tree.

Trees, homology, and automorphism groups of RAAGs

Abstract

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this subgroup, based on the number and degree of a certain type of vertices, which we call deep. We then use combinatorial methods to analyze the average value of this Betti number, in terms of the size of the defining tree.

Paper Structure

This paper contains 20 sections, 17 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Abelianization of finite-index subgroups
  3. Graphs and trees
  4. Trees
  5. Automorphisms of RAAGs defined by trees
  6. Combinatorics of deep nodes
  7. Plan of the paper
  8. RAAGs and their automorphisms
  9. A partial order on the set of nodes
  10. Laurence--Servatius generators
  11. Day's presentation of $\mathop{\mathrm{Aut}}\nolimits(A_K)$
  12. The group ${\mathop{\mathrm{Aut}}\nolimits^\star}(A_K)$
  13. Proof of Theorem \ref{['mainthm:hom']}
  14. A remark on the bound given by Theorem \ref{['mainthm:hom']}
  15. Deep nodes and shallow trees
  16. ...and 5 more sections

Key Result

Theorem 1

If the tree $T$ is not shallow, then $b_1({\mathop{\mathrm{Aut}}\nolimits^\star}(A_T)) \ge 1$.

Figures (1)

  • Figure 1: Trees with $\partial_{\rm root}\ge 3$.

Theorems & Definitions (27)

  • Theorem : AMP
  • Theorem A
  • Theorem B
  • Lemma 2.1
  • Theorem 2.2: Laurence, Servatius
  • Lemma 2.3: Day, Lemma 2.5
  • Remark 2.4
  • Theorem 2.5: Day
  • Remark 2.6
  • Proposition 2.7
  • ...and 17 more