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Graphs and complexes of lattices

Sam Hughes

TL;DR

This work develops a comprehensive framework of graphs and complexes of lattices to study lattices in products of CAT(0) spaces via commensurated subgroups. It proves a general structure theorem linking uniform (H×T)-lattices to finite graphs of H-lattices (and a complex-analogue), and then derives criteria for C*-simplicity, including irreducibility–faithfulness equivalences in Euclidean-product settings. The paper furthermore constructs non-residually finite and non-biautomatic lattices using a functorial transition from graphs of groups to complexes of groups (via Thomas’ construction) and extends actions to Salvetti complexes, yielding towers of lattices with vanishing covolume. It also provides concrete examples in products involving right-angled buildings and RAAGs, and discusses the limits and questions arising from these methods. Overall, the results advance the understanding of lattice structure, rigidity, and operator-algebraic properties for CAT(0) groups in complex product geometries.

Abstract

We study lattices acting on $\mathrm{CAT}(0)$ spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices and a complex of lattices giving graph and complex of group splittings of $\mathrm{CAT}(0)$ lattices. Using this framework we characterise irreducible uniform $(\mathrm{Isom}(\mathbb{E}^n)\times T)$-lattices by $C^\ast$-simplicity and give a necessary condition for lattices in products with a Euclidean factor to be biautomatic. We also construct non-residually finite uniform lattices acting on arbitrary products of right-angled buildings and non-biautomatic lattices acting on the product of $\mathbb{E}^n$ and a right-angled building.

Graphs and complexes of lattices

TL;DR

This work develops a comprehensive framework of graphs and complexes of lattices to study lattices in products of CAT(0) spaces via commensurated subgroups. It proves a general structure theorem linking uniform (H×T)-lattices to finite graphs of H-lattices (and a complex-analogue), and then derives criteria for C*-simplicity, including irreducibility–faithfulness equivalences in Euclidean-product settings. The paper furthermore constructs non-residually finite and non-biautomatic lattices using a functorial transition from graphs of groups to complexes of groups (via Thomas’ construction) and extends actions to Salvetti complexes, yielding towers of lattices with vanishing covolume. It also provides concrete examples in products involving right-angled buildings and RAAGs, and discusses the limits and questions arising from these methods. Overall, the results advance the understanding of lattice structure, rigidity, and operator-algebraic properties for CAT(0) groups in complex product geometries.

Abstract

We study lattices acting on spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices and a complex of lattices giving graph and complex of group splittings of lattices. Using this framework we characterise irreducible uniform -lattices by -simplicity and give a necessary condition for lattices in products with a Euclidean factor to be biautomatic. We also construct non-residually finite uniform lattices acting on arbitrary products of right-angled buildings and non-biautomatic lattices acting on the product of and a right-angled building.

Paper Structure

This paper contains 27 sections, 33 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices and covolumes
  4. Non-positive curvature
  5. Irreducibility
  6. Graphs of lattices
  7. Graphs of groups
  8. A Structure theorem
  9. Reducible lattices
  10. Examples
  11. An application to C*-simplicity
  12. Some conditions for C*-simplicity
  13. C*-simplicity of (HxT)-lattices
  14. Faithful actions of Isom(En)xT)-lattices
  15. C*-simplicity of (Isom(En)xT)-lattices
  16. ...and 12 more sections

Key Result

Theorem B

Let $\mathcal{T}$ be a locally-finite leafless tree and $T=\mathop{\mathrm{\mathrm{Aut}}}\nolimits(\mathcal{T})$ be unimodular. Let $n\geq1$ and $\Gamma<\mathop{\mathrm{\mathrm{Isom}}}\nolimits(\mathbb{E}^n)\times T$ be a uniform lattice. The following are equivalent:

Figures (1)

  • Figure 1: The left pentagon shows a labelling of the types $J\in{\mathcal{S}}$. The right pentagon shows the local groups after applying Thomas' functor to a graph of groups with a single edge. In both pentagons the dashed line shows the embedding of the graph. If the graph of groups has a single vertex, then $G_v=G_w$, $q_1=q_2$, $q_3=q_4$, the edge $(\{i_1,i_5\},\{i_1\})$ is glued to $(\{i_2,i_5\},\{i_2\})$, and the edge $(\{i_1,i_3\},\{i_1\})$ is glued to $(\{i_2,i_4\},\{i_2\})$.

Theorems & Definitions (74)

  • Remark 1.2
  • Definition 1.3
  • Theorem B
  • Corollary C: Special case of Corollary \ref{['cor.thomas.notbiaut']}
  • Corollary D: Special case of Corollary \ref{['thm.treelatstobuildings']}
  • Theorem 2.1: Serre's covolume formula Serre1971
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • ...and 64 more