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arxiv: 2605.28963 · v1 · pith:D6F4WNP2new · submitted 2026-05-27 · 🧮 math.GR · math.AT· math.GT· math.MG

Generalisable presentations and compactness properties of locally compact right-angled Artin groups

Ilaria Castellano , Bianca Marchionna , Brita Nucinkis , Yuri Santos Rego This is my paper

Pith reviewed 2026-06-29 09:11 UTC · model grok-4.3

classification 🧮 math.GR math.ATmath.GTmath.MG
keywords right-angled Artin groupslocally compact groupsgeneralisable presentationsSalvetti complexesfiniteness propertiescohomological dimensionBieri-Stallings constructiontopological groups
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The pith

Right-angled Artin groups admit generalisable presentations that produce locally compact groups containing them as discrete subgroups with controlled compactness and finiteness properties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes presentations that can be generalised over a continuous open group monomorphism, turning discrete presentations into those of topological groups with a prescribed open subgroup. It applies this method to right-angled Artin groups to define topological RAAGs as locally compact groups that contain the original RAAG as a discrete subgroup. The authors introduce and study universal Salvetti-type complexes for these groups, establishing high connectivity for a large class of examples. This produces new locally compact groups with prescribed compactness properties or rational cohomological dimension and extends the Bieri-Stallings construction to obtain TDLC groups of type FP_n but not FP_{n+1}, along with new discrete groups of type F_n but not F_{n+1}.

Core claim

Generisable presentations applied to right-angled Artin groups yield locally compact topological RAAGs. For many examples the associated universal Salvetti-type complexes are highly connected, which gives LC groups with prescribed compactness properties or rational cohomological dimension. The authors extend the Bieri-Stallings construction to obtain TDLC groups of type FP_n but not FP_{n+1} and record counterparts of cohomological results such as a Mayer-Vietoris sequence and Kunneth formula in discrete cohomology for TDLC groups. As a by-product they obtain new discrete groups with controlled finiteness properties, including Thompson-like Bieri-Stallings groups of type F_n but not F_{n+1}

What carries the argument

Generalisable presentations over continuous open group monomorphisms, which embed discrete group presentations into topological groups with a prescribed open subgroup; applied here to RAAGs to form topological RAAGs whose universal Salvetti-type complexes carry the geometric and connectivity arguments.

If this is right

  • LC groups exist with prescribed compactness properties or rational cohomological dimension.
  • TDLC groups exist of type FP_n but not FP_{n+1}.
  • Discrete groups exist of type F_n but not F_{n+1}, including Thompson-like Bieri-Stallings examples for every n.
  • Mayer-Vietoris sequences and Kunneth formulas hold in discrete cohomology for TDLC groups.
  • Universal Salvetti-type complexes provide models for classifying spaces in some cases even when they are not CAT(0) cube complexes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be applied directly to the orientable surface groups, Artin groups, and Thompson groups mentioned to produce additional families of topological groups with controlled properties.
  • High connectivity of the complexes might allow explicit constructions of models for classifying spaces in the locally compact setting beyond the cases where they are CAT(0).
  • The distinction from lattice envelopes suggests these groups can be used to study actions where the discrete subgroup is not cocompact.
  • The non-uniquely geodesic cases indicate that these complexes can serve as test objects for geometric properties that differ from standard cube complexes.

Load-bearing premise

The generalisable presentations for RAAGs produce locally compact groups whose universal Salvetti-type complexes satisfy the high-connectivity properties needed to reach the stated compactness and finiteness conclusions.

What would settle it

A concrete RAAG example where the associated universal Salvetti-type complex fails to be highly connected or where the resulting topological group does not have type FP_n but not FP_{n+1}.

Figures

Figures reproduced from arXiv: 2605.28963 by Bianca Marchionna, Brita Nucinkis, Ilaria Castellano, Yuri Santos Rego.

Figure 1
Figure 1. Figure 1: SrΓpφq for Γ “ ‚—‚ and φ: z P Z2 ÞÑ 2z P Z2. For more information and a 3D visualisation of this model, see Raphael Appenzeller’s Thingiverse page: https://www.thingiverse.com/thing:7360375. (ii) On the other extreme, assume that Γ is totally disconnected, i.e., Γ “ pS, ∅q, and U is Hausdorff. In this case, AΓpφq “ HSpφq is the iterated HNN-extension over φ with stable letters S. By design SrΓpφq is 1-dime… view at source ↗
Figure 2
Figure 2. Figure 2: A pocket in a generalised universal Salvetti complex. The presence of pockets in SrΓpφq has the following consequences [PITH_FULL_IMAGE:figures/full_fig_p051_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distinct geodesics γ1, γ2 with same endpoints. (iii) The space SrΓpφq, endowed with the piecewise ℓ 8-metric from its squares, is not injective [Lan13, Section 2]. Indeed, consider two orthogonal medians in each of Q and gQ, and select three closed balls B1, B2, B3 as in [PITH_FULL_IMAGE:figures/full_fig_p052_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Closed balls B1, B2, B3 without the Helly property. Then Bi X Bj ‰ ∅ for all 1 ď i ă j ď 3, but B1 X B2 X B3 “ ∅. In view of Example 5.7, are there mild conditions under which SrΓpφq is highly connected or in fact a CATp0q cube complex? Of course, there are trivial answers to these questions as seen earlier: SrΓpφq is a CATp0q cube complex when U “ t1u or when Γ is totally disconnected; see Example 5.3. We… view at source ↗
read the original abstract

We propose the systematic study of presentations that can be generalised over a continuous open group monomorphism. Presentations with this property can turn well-known presentations such as those for as orientable surface groups, Artin groups, and some Thompson groups, into topological groups with a prescribed open subgroup. Later we focus on right-angled Artin groups (RAAGs) and introduce a notion of topological RAAGs. Our approach differs from lattice envelopes and produces examples of locally compact (LC) groups that contain RAAGs as discrete subgroups, but generally not as lattices. We investigate some geometric aspects of topological RAAGs, with a special emphasis on compactness properties of LC ones. This includes a study of universal Salvetti-type complexes which may be of independent interest. These complexes share some properties with buildings. Although in some cases they are CAT(0) cube complexes and provide models for classifying spaces, in other cases they are not even uniquely geodesic. For a large class of examples we establish high connectivity properties for these complexes. This yields novel examples of LC groups with prescribed compactness properties or rational cohomological dimension. We note that the Bestvina-Brady machinery does not automatically generalise to this setting; nevertheless, we extend the Bieri-Stallings construction to obtain totally disconnected locally compact (TDLC) groups of type $FP_n$ but not $FP_{n+1}$. Along the way we record counterparts of cohomological results, such as a Mayer-Vietoris sequence and K\"unneth formula in discrete (co)homology for TDLC groups, which have not appeared elsewhere in the literature. Despite our non-discrete LC focus we obtain, as by-product, new examples of discrete groups with controlled finiteness properties including, for every $n \geq 1$, a Thompson-like Bieri-Stallings group of type $F_n$ but not $F_{n+1}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proposes the systematic study of 'generalisable presentations' that extend over continuous open group monomorphisms, applies this framework to right-angled Artin groups to define topological RAAGs (distinct from lattice envelopes), investigates geometric aspects of these groups with emphasis on compactness properties via universal Salvetti-type complexes (which share properties with buildings, are sometimes CAT(0) cube complexes, and may model classifying spaces), establishes high connectivity of these complexes for a large class of examples yielding LC groups with prescribed compactness or rational cohomological dimension, extends the Bieri-Stallings construction to produce TDLC groups of type FP_n but not FP_{n+1}, records Mayer-Vietoris and Künneth formulas for discrete (co)homology of TDLC groups, and obtains new discrete groups (including Thompson-like examples) with controlled finiteness properties.

Significance. If the constructions and proofs are correct, the work supplies novel examples of locally compact groups containing RAAGs as discrete (but typically non-lattice) subgroups together with explicit control over compactness and cohomological dimension; the extension of Bieri-Stallings to the TDLC setting and the new cohomological sequences for TDLC groups are substantive additions to the literature on finiteness properties beyond the discrete case.

minor comments (3)
  1. [Abstract] Abstract, paragraph on topological RAAGs: the phrase 'for a large class of examples' is used for the high-connectivity result; the precise class (e.g., a stated condition on the defining graph or the monomorphism) should be recalled explicitly in the introduction or the relevant theorem statement.
  2. [Introduction / §5] The claim that the Bestvina-Brady machinery 'does not automatically generalise' is stated without a concrete counter-example or obstruction; a brief illustration in §4 or §5 would clarify the necessity of the new Bieri-Stallings extension.
  3. [Cohomology section] The paper records Mayer-Vietoris and Künneth formulas for discrete (co)homology of TDLC groups; these appear as new results, so a short comparison with existing literature on TDLC cohomology (even if none exist) would strengthen the 'have not appeared elsewhere' assertion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Derivation chain is self-contained; no reductions by construction

full rationale

The paper defines generalisable presentations and topological RAAGs as new objects, then derives high-connectivity results for their Salvetti-type complexes and extends the Bieri-Stallings construction to obtain TDLC groups of type FP_n but not FP_{n+1}. These steps are presented as direct consequences of the introduced definitions and standard topological group theory, with no equations or claims that reduce to fitted inputs, self-citations, or prior results by the same authors. The abstract and described content contain no self-definitional loops, renamed empirical patterns, or load-bearing self-citations that would force the conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the newly introduced definitions of generalisable presentations and topological RAAGs plus standard background facts from geometric group theory; no free parameters or invented physical entities are present.

axioms (2)
  • domain assumption Continuous open group monomorphisms preserve the structure needed to extend discrete presentations to topological groups
    Invoked when turning well-known presentations into topological groups with prescribed open subgroups.
  • ad hoc to paper Universal Salvetti-type complexes can be defined and studied for the new topological RAAGs
    Central to the geometric analysis and connectivity claims.
invented entities (1)
  • topological RAAG no independent evidence
    purpose: Locally compact group containing a discrete RAAG as subgroup but generally not as lattice
    New object defined in the paper to produce the claimed examples.

pith-pipeline@v0.9.1-grok · 5906 in / 1462 out tokens · 35095 ms · 2026-06-29T09:11:05.068358+00:00 · methodology

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