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Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one

Ilaria Castellano, Bianca Marchionna, Thomas Weigel

TL;DR

The paper extends the Stallings–Swan paradigm to locally compact, totally disconnected groups by proving that every compactly generated, $\mathcal{CO}$-bounded t.d.l.c. group $G$ with rational discrete cohomological dimension $\mathrm{cd}_{\mathbb{Q}}(G)\leq 1$ is the fundamental group of a finite graph of profinite groups, generalizing Dunwoody’s rational result. Central to the approach is the development of rational discrete cohomology for t.d.l.c. groups, the associated $W^*$-algebras via Hecke pairs, and a trace framework that yields non-positive Euler–Poincaré characteristics for unimodular groups with $\mathrm{cd}_{\mathbb{Q}}(G)=1$. The paper then establishes accessibility results, showing that under $\mathcal{CO}$-boundedness (or compact presentability) and unimodularity, such groups are decomposable as finite graphs of profinite groups, with several equivalent characterizations in terms of Bass–Serre theory and Cayley–Abels graphs. Collectively, these results provide a structural theory for a broad class of t.d.l.c. groups, connecting cohomological, analytic, and combinatorial perspectives and extending classical Stallings–Swan theory to the topological setting.

Abstract

It is shown that a Stallings--Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Thm. B). More precisely, a compactly generated $\mathcal{CO}$-bounded t.d.l.c. group $G$ of rational discrete cohomological dimension less than or equal to $1$ must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody's rational version of the classical Stallings--Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension $1$ has necessarily non-positive Euler--Poincaré characteristic (cf. Thm. H).

Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one

TL;DR

The paper extends the Stallings–Swan paradigm to locally compact, totally disconnected groups by proving that every compactly generated, -bounded t.d.l.c. group with rational discrete cohomological dimension is the fundamental group of a finite graph of profinite groups, generalizing Dunwoody’s rational result. Central to the approach is the development of rational discrete cohomology for t.d.l.c. groups, the associated -algebras via Hecke pairs, and a trace framework that yields non-positive Euler–Poincaré characteristics for unimodular groups with . The paper then establishes accessibility results, showing that under -boundedness (or compact presentability) and unimodularity, such groups are decomposable as finite graphs of profinite groups, with several equivalent characterizations in terms of Bass–Serre theory and Cayley–Abels graphs. Collectively, these results provide a structural theory for a broad class of t.d.l.c. groups, connecting cohomological, analytic, and combinatorial perspectives and extending classical Stallings–Swan theory to the topological setting.

Abstract

It is shown that a Stallings--Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Thm. B). More precisely, a compactly generated -bounded t.d.l.c. group of rational discrete cohomological dimension less than or equal to must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody's rational version of the classical Stallings--Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension has necessarily non-positive Euler--Poincaré characteristic (cf. Thm. H).
Paper Structure (15 sections, 23 theorems, 70 equations)

This paper contains 15 sections, 23 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Rational discrete cohomology: a summary
  3. The $W^*$-algebra of a Hecke pair
  4. $C^*$-algebras
  5. $W^*$-algebras
  6. The algebra $\mathcal{H}(G,\mathcal{O})_{\mathbb{Q}}$
  7. $\mathcal{H}(G,\mathcal{O})_{\mathbb{C}}$ as operator algebra
  8. The trace map on $M_n(W(G,\mathcal{O}))$
  9. Euler--Poincaré characteristic of unimodular t.d.l.c. groups with $\mathrm{cd}_{\mathbb{Q}}(G)\leq 1$
  10. The Hattori--Stallings rank of discrete ${\mathbb{Q}}[G]$-modules of type $\mathrm{FP}$
  11. Discrete left augmentation ${\mathbb{Q}}[G]$-modules
  12. Accessibility: general results
  13. Graphs, trees and graphs of groups
  14. Accessibility of t.d.l.c. groups
  15. $\mathcal{CO}$-bounded t.d.l.c. groups

Key Result

Proposition 3.4

The ${\mathbb{C}}$-linear map can be extended to the ${\mathbb{C}}$-linear map $\tau\colon W(G,\mathcal{O})\longrightarrow {\mathbb{C}}$ given by which is a positive and faithful trace with $\tau(F^*)=\overline{\tau(F)}$ for all $F\in W(G,O)$.

Theorems & Definitions (52)

  • proof
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • proof
  • ...and 42 more