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arxiv: 2604.05123 · v1 · submitted 2026-04-06 · 🧮 math.GR

Lattices determined by their commensurator

Adrien Le Boudec , Colin Reid This is my paper

Pith reviewed 2026-05-10 18:41 UTC · model grok-4.3

classification 🧮 math.GR
keywords latticescommensuratorstotally disconnected locally compact groupstree automorphismsrigiditycocompact latticesgraph products
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The pith

Every finitely generated commensurated subgroup of the commensurator is virtually contained in the lattice.

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

The paper examines finitely generated commensurated subgroups inside the commensurator C of a lattice Γ in a totally disconnected locally compact group G. It establishes rigidity results showing that every such subgroup must be virtually contained in Γ. In concrete settings such as automorphism groups of trees, this implies that Γ is the only infinite finitely generated commensurated subgroup up to commensurability. The results settle questions about whether distinct non-commensurable lattices can share a commensurator and extend to graph products in right-angled buildings.

Core claim

Let Γ be a finitely generated cocompact lattice of a totally disconnected locally compact group G, and C a dense subgroup of G that contains and commensurates Γ. Every finitely generated commensurated subgroup of C is virtually contained in Γ. In more concrete situations such as when G is the automorphism group of a tree, Γ is the only infinite finitely generated commensurated subgroup of C up to commensurability.

What carries the argument

The commensurator C of Γ in G, with the finite generation and cocompactness of Γ in the totally disconnected locally compact group G, which together produce the rigidity of commensurated subgroups.

If this is right

  • Two non-commensurable cocompact lattices in the automorphism group of a tree cannot share the same commensurator.
  • The same uniqueness holds for commensurators of cocompact lattices in other groups of automorphisms of trees.
  • Commensurators of graph products of finite groups in automorphism groups of right-angled buildings satisfy the same rigidity.

Where Pith is reading between the lines

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

  • The commensurator may be viewed as an object largely fixed by its lattice, which could constrain constructions of exotic subgroups in these groups.
  • Similar uniqueness statements might be tested computationally in specific low-degree tree lattices or right-angled buildings.
  • The results connect to broader questions about how much algebraic structure is preserved when passing from a lattice to its commensurator.

Load-bearing premise

That Γ is finitely generated and G is totally disconnected and locally compact.

What would settle it

An explicit example of a totally disconnected locally compact group G with a finitely generated cocompact lattice Γ and dense commensurating subgroup C that contains an infinite finitely generated commensurated subgroup not virtually contained in Γ.

read the original abstract

Let $\Gamma$ be a finitely generated cocompact lattice of a totally disconnected locally compact group $G$, and $C$ a dense subgroup of $G$ that contains and commensurates $\Gamma$. We study the problem of describing all finitely generated commensurated subgroups of $C$. We establish general rigidity results ensuring every finitely generated commensurated subgroup of $C$ is virtually contained in $\Gamma$. In more concrete situations, in fact we conclude that up to commensurability, $\Gamma$ is the only infinite finitely generated commensurated subgroup of $C$. For instance this last conclusion holds when $G$ is the automorphism group of a tree. This settles in particular the problem whether two non-commensurable cocompact tree lattices may have the same commensurator. Further applications include commensurators of cocompact lattices in other groups of automorphisms of trees, as well as commensurators of graph product of finite groups in automorphism groups of right-angled building.

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 / 2 minor

Summary. The paper claims that if Γ is a finitely generated cocompact lattice in a totally disconnected locally compact group G and C is a dense subgroup of G containing and commensurating Γ, then every finitely generated commensurated subgroup of C is virtually contained in Γ. In concrete cases (e.g., G the automorphism group of a tree), Γ is the unique infinite f.g. commensurated subgroup of C up to commensurability. This settles whether non-commensurable cocompact tree lattices can share a commensurator, with further applications to other tree automorphism groups and commensurators of graph products of finite groups in right-angled buildings.

Significance. If the results hold, they deliver strong rigidity theorems for commensurated subgroups in td lc groups, resolving a concrete open question on uniqueness of tree lattices with a given commensurator. The general framework leads to specific conclusions via standard group-theoretic constructions without free parameters or ad-hoc entities, and the applications to Aut(T) and right-angled buildings provide falsifiable predictions in geometric group theory.

minor comments (2)
  1. [§1] §1 (Introduction): the statement of the main theorem could explicitly restate the finite-generation and cocompactness hypotheses on Γ immediately before the rigidity conclusion for clarity.
  2. [§4] §4 (Applications to trees): the argument that the conclusion holds for G = Aut(T) relies on a prior result about cocompact lattices; a one-sentence pointer to the exact citation would help readers trace the reduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report, which accurately summarizes the main results and highlights the significance of the rigidity theorems for commensurated subgroups. We appreciate the recommendation for minor revision and will incorporate any editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents pure mathematical theorems establishing rigidity for finitely generated commensurated subgroups of a dense subgroup C containing a cocompact lattice Γ in a td lc group G. Derivations rely on standard group-theoretic arguments (e.g., properties of automorphism groups of trees, commensurability, virtual containment) under explicit hypotheses of finite generation and td lc structure. No statistical fits, predictions of fitted quantities, self-definitional reductions, or load-bearing self-citation chains appear; any prior citations support independent results rather than closing a loop back to the current claims. The central conclusions follow from the stated assumptions without reducing to them by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

From the abstract alone, the paper relies on standard domain assumptions in the theory of lattices and commensurators in td lc groups; no free parameters or new entities are introduced in the summary.

axioms (2)
  • domain assumption Γ is a finitely generated cocompact lattice in a totally disconnected locally compact group G
    This is the basic setup for the problem as stated in the abstract.
  • domain assumption C is a dense subgroup of G containing and commensurating Γ
    Central definition for the commensurator and the subgroups studied.

pith-pipeline@v0.9.0 · 5461 in / 1186 out tokens · 40774 ms · 2026-05-10T18:41:58.676195+00:00 · methodology

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Reference graph

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