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Subgroups of Bestvina-Brady groups

Simone Blumer

TL;DR

This work characterizes exactly when every subgroup of a Bestvina-Brady group B_Gamma is a RAAG, revealing that this holds iff Gamma is chordal and avoids two specific induced subgraphs (gem and overlapping-gems). It then develops a graph-theoretic framework via ptolemaic graphs and trees of Droms graphs to translate algebraic properties into combinatorial conditions, and extends the theory to Lie algebras and pro-p completions, including cohomological and Koszulity aspects. The results exclude high-genus surface groups from Bestvina-Brady realizations and provide Bloch-Kato type and May spectral sequence tools to study the cohomology and finiteness properties of these groups and their Lie algebra analogues. The findings unify coherence, local-RAAG behavior, and pro-p Galois-analytic conjectures within the Bestvina-Brady setting, with concrete graph-theoretic criteria and explicit inequalities for acyclic flag complexes.

Abstract

In "Subgroups of Graph Groups", 1987, J. Alg., Droms proved that all the subgroups of a right-angled Artin group (RAAG) defined by a finite simplicial graph $Γ$ are themselves RAAGs if, and only if, $Γ$ has no induced square graph nor line-graph of length $3$. The present work provides a similar result for specific normal subgroups of RAAGs, called Bestvina-Brady groups: We characterize those graphs in which every subgroup of such a group is itself a RAAG. In turn, we confirm several Galois theoretic conjectures for the pro-$p$ completions of these groups.

Subgroups of Bestvina-Brady groups

TL;DR

This work characterizes exactly when every subgroup of a Bestvina-Brady group B_Gamma is a RAAG, revealing that this holds iff Gamma is chordal and avoids two specific induced subgraphs (gem and overlapping-gems). It then develops a graph-theoretic framework via ptolemaic graphs and trees of Droms graphs to translate algebraic properties into combinatorial conditions, and extends the theory to Lie algebras and pro-p completions, including cohomological and Koszulity aspects. The results exclude high-genus surface groups from Bestvina-Brady realizations and provide Bloch-Kato type and May spectral sequence tools to study the cohomology and finiteness properties of these groups and their Lie algebra analogues. The findings unify coherence, local-RAAG behavior, and pro-p Galois-analytic conjectures within the Bestvina-Brady setting, with concrete graph-theoretic criteria and explicit inequalities for acyclic flag complexes.

Abstract

In "Subgroups of Graph Groups", 1987, J. Alg., Droms proved that all the subgroups of a right-angled Artin group (RAAG) defined by a finite simplicial graph are themselves RAAGs if, and only if, has no induced square graph nor line-graph of length . The present work provides a similar result for specific normal subgroups of RAAGs, called Bestvina-Brady groups: We characterize those graphs in which every subgroup of such a group is itself a RAAG. In turn, we confirm several Galois theoretic conjectures for the pro- completions of these groups.

Paper Structure

This paper contains 11 sections, 21 theorems, 16 equations, 3 figures.

Table of Contents

  1. Right-angled Artin & Bestvina-Brady groups and Lie algebras
  2. RAAGs
  3. Bestvina-Brady
  4. Lower central series
  5. Graph characterization
  6. Ptolemaic graphs
  7. The defining construction of trees of Droms graphs
  8. Applications of Theorem \ref{['thm:bkbb']}
  9. Surface groups
  10. Bloch-Kato version of the $b_2$-conjecture
  11. Coherence of Bestvina-Brady objects

Key Result

Theorem 1

Let $\Gamma$ be a finite graph. Then, every subgroup of $G_\Gamma$ is a RAAG if, and only if, no induced subgraph of $\Gamma$ is either a square or a line-graph of length $3$.

Figures (3)

  • Figure 1: The overlapping-gems graph $\bar{H}$ (in black) and the defining graph of $\mathfrak{B}_{\bar{H}}$ (in blue).
  • Figure :
  • Figure :

Theorems & Definitions (36)

  • Theorem 1: Droms
  • Theorem 2: Snopče, Zalesskii ilirPavel
  • Proposition 3
  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 1.7
  • ...and 26 more