Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Irreducible lattices fibring over the circle

Sam Hughes

TL;DR

The work develops a detailed study of BNSR (Sigma) invariants for irreducible uniform lattices in products of CAT(0) spaces, linking irreducibility to fibering properties and finiteness properties of kernel subgroups. It combines CAT(0) geometry, group cohomology, and graph-of-groups techniques with constructions that lift lattices from trees to Salvetti complexes, producing irreducible lattices that fiber over the circle in controlled ways. A key result establishes that for lattices in $\mathrm{Isom}(\mathbb{E}^n)\times T$, irreducibility is equivalent to the nonexistence of a virtual $\mathsf{F}_1$-fibering, while a complementary construction yields irreducible uniform lattices that do fiber over $S^1$, with finiteness properties of kernels governed by Bestvina–Brady-type relationships via RAAGs. The work also develops a transfer principle between finiteness properties of kernels in RAAGs and in the corresponding lifted lattice $\Gamma_L$, providing new examples of irreducible lattices with prescribed fibering behavior and finiteness profiles. Overall, the results illuminate the interplay between irreducibility, fibering invariants $\Sigma^n$, and finiteness properties in higher-rank CAT(0) lattice theory, with implications for constructing and understanding nontrivial fibrations in this geometric group theory setting.

Abstract

We investigate the Bieri--Neumann--Strebel--Renz (BNSR) invariants of irreducible uniform lattices. In the case of a direct product of a tree and a Euclidean space we show that vanishing of the BNSR invariants for all finite-index subgroups of a given uniform lattice is equivalent to irreducibility. On the other hand we construct irreducible uniform lattices which admit maps to the integers whose kernels' finiteness properties are determined by the finiteness properties of certain Bestvina--Brady groups.

Irreducible lattices fibring over the circle

TL;DR

The work develops a detailed study of BNSR (Sigma) invariants for irreducible uniform lattices in products of CAT(0) spaces, linking irreducibility to fibering properties and finiteness properties of kernel subgroups. It combines CAT(0) geometry, group cohomology, and graph-of-groups techniques with constructions that lift lattices from trees to Salvetti complexes, producing irreducible lattices that fiber over the circle in controlled ways. A key result establishes that for lattices in , irreducibility is equivalent to the nonexistence of a virtual -fibering, while a complementary construction yields irreducible uniform lattices that do fiber over , with finiteness properties of kernels governed by Bestvina–Brady-type relationships via RAAGs. The work also develops a transfer principle between finiteness properties of kernels in RAAGs and in the corresponding lifted lattice , providing new examples of irreducible lattices with prescribed fibering behavior and finiteness profiles. Overall, the results illuminate the interplay between irreducibility, fibering invariants , and finiteness properties in higher-rank CAT(0) lattice theory, with implications for constructing and understanding nontrivial fibrations in this geometric group theory setting.

Abstract

We investigate the Bieri--Neumann--Strebel--Renz (BNSR) invariants of irreducible uniform lattices. In the case of a direct product of a tree and a Euclidean space we show that vanishing of the BNSR invariants for all finite-index subgroups of a given uniform lattice is equivalent to irreducibility. On the other hand we construct irreducible uniform lattices which admit maps to the integers whose kernels' finiteness properties are determined by the finiteness properties of certain Bestvina--Brady groups.
Paper Structure (18 sections, 17 theorems, 6 equations, 1 figure)

This paper contains 18 sections, 17 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conjectures and questions
  3. Acknowledgements
  4. Bibliographical note
  5. Preliminaries
  6. Lattices
  7. Irreducibility
  8. Graphs and complexes of lattices
  9. Leary--Minasyan groups
  10. Fibring lattices in a product of a tree and a Euclidean space
  11. Lattices in symmetric spaces and Salvetti complexes
  12. Right-angled Artin groups
  13. The plan
  14. From a 3-valent to a 4-valent tree
  15. From a 4-valent tree to a Salvetti complex
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a lattice in semisimple Lie group with finite centre and real rank at least $2$. If $H^1(\Gamma)\neq0$, then $\Gamma$ virtually splits as a direct product of two infinite groups.

Figures (1)

  • Figure 1: An illustration of Jingyin Huang's trick to turn a $3$-regular tree into a $4$-regular graph.

Theorems & Definitions (33)

  • Theorem 1.1: Margulis
  • Conjecture 1.3
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof : Proof of \ref{['prop.isomEnxX.H1']}
  • Claim 3.2
  • proof : Proof of claim:
  • Proposition 3.3: Cashen--Levitt
  • Theorem 1
  • ...and 23 more