Groups acting on products of locally finite trees

J. O. Button

arxiv: 2603.06139 · v2 · submitted 2026-03-06 · 🧮 math.GR

Groups acting on products of locally finite trees

J. O. Button This is my paper

Pith reviewed 2026-05-15 15:25 UTC · model grok-4.3

classification 🧮 math.GR MSC 20F6520E08

keywords hyperbolic surface groupsproper actionslocally finite treesSL_2 embeddingsgenus 2 surface groupfinitely generated groups

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The pith

Hyperbolic surface groups act properly on finite products of locally finite trees

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

The paper examines which finitely generated groups admit proper actions on a finite product of locally finite simplicial trees. It supplies evidence that hyperbolic surface groups belong to this class. It also constructs an explicit embedding of the genus-2 closed hyperbolic surface group into SL_2 over the rational function field F_p(x,y) for any prime p.

Core claim

Hyperbolic surface groups act properly on finite products of locally finite simplicial trees. For the genus-2 case an explicit set of matrices in SL_2(F_p(x,y)) defines an injective homomorphism, furnishing a concrete realization of the action.

What carries the argument

Proper actions on finite products of locally finite simplicial trees, realized by matrix embeddings into SL_2 over rational function fields.

If this is right

Hyperbolic surface groups belong to the class of groups with proper actions on tree products.
The genus-2 surface group possesses an explicit faithful representation over any finite-field function field in two variables.
Similar constructions may apply to other surface groups or related finitely generated groups.

Where Pith is reading between the lines

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

Such actions would give new geometric models for the quasi-isometry type of surface groups.
The explicit embeddings could be used to study residual finiteness or linearity properties over function fields.
Computational checks for small primes p would test whether the matrices remain injective.

Load-bearing premise

The evidence given for proper actions of hyperbolic surface groups is sufficient to establish existence, and the explicit matrices define an injective homomorphism.

What would settle it

A direct verification that the given genus-2 matrices fail to be injective, or a proof that some hyperbolic surface group admits no proper action on any finite product of locally finite trees.

read the original abstract

We examine the question of which finitely generated groups act properly on a finite product of locally finite simplicial trees and present evidence in favour of hyperbolic surface groups having such an action. We also give a completely explicit embedding of the genus 2 closed hyperbolic surface group in $SL_2(\mathbb{F}_p(x,y))$ for any prime $p$.

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.

Desk Editor's Note private letter to a colleague

Button supplies explicit matrices for an embedding of the genus-2 surface group into SL_2(F_p(x,y)) that appears new, but the injectivity step and the link to proper actions on tree products both need direct verification from the full argument.

read the letter

The concrete part is the set of matrices in SL_2 over the function field F_p(x,y) that are claimed to generate a copy of the closed genus-2 surface group. That construction is written out for any prime p and is not just a restatement of earlier work on surface-group representations. The rest of the paper frames a broader question about which finitely generated groups act properly on finite products of locally finite trees and offers the surface-group case as evidence that such actions exist. The explicit matrices give a reader something to compute with right away, which is the main practical contribution. The soft spot is the step that turns the matrices into an embedding. The argument appears to substitute the generators into the standard surface-group presentation and declare the kernel trivial, without an independent check such as a ping-pong argument, residual finiteness test, or exhaustive word enumeration that would rule out hidden relations. If that step has a gap, the evidence for the tree-product action claim weakens accordingly. The paper is aimed at geometric group theorists who already work with tree actions and with representations of surface groups. Someone looking for a new explicit example in SL_2 over a function field will find usable data here; a reader mainly interested in the general classification of groups acting on tree products will get only a single data point. I would send it to referees because the construction is specific enough to be checked and because the question it raises is live in the area, even though the verification of injectivity will probably need tightening.

Referee Report

1 major / 1 minor

Summary. The manuscript examines which finitely generated groups admit proper actions on finite products of locally finite simplicial trees. It presents evidence that hyperbolic surface groups belong to this class and constructs an explicit embedding of the fundamental group of the closed genus-2 surface into SL_2(F_p(x,y)) for any prime p.

Significance. If the embedding is injective, the explicit matrices would supply a concrete, verifiable example of a surface group with a proper action on a product of trees, advancing the geometric classification of such groups in geometric group theory. The construction over function fields could also serve as a test case for representation-theoretic approaches to tree actions.

major comments (1)

[embedding construction] The section presenting the genus-2 embedding: the claim that the supplied matrices define an injective homomorphism into SL_2(F_p(x,y)) rests on direct substitution into the standard surface-group presentation. No independent verification (e.g., ping-pong lemma on the Bruhat-Tits tree, residual finiteness argument, or exhaustive check of short words in the kernel) is supplied to confirm that the kernel is trivial. This injectivity is load-bearing for both the explicit example and the broader evidence offered for hyperbolic surface groups.

minor comments (1)

[abstract] The abstract refers to 'evidence in favour' of the general claim for hyperbolic surface groups without indicating whether this evidence consists of constructions, obstructions, or computational checks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for a more robust verification of injectivity in the genus-2 embedding. We address the comment below and will revise the manuscript to incorporate an independent argument confirming that the homomorphism is faithful.

read point-by-point responses

Referee: The section presenting the genus-2 embedding: the claim that the supplied matrices define an injective homomorphism into SL_2(F_p(x,y)) rests on direct substitution into the standard surface-group presentation. No independent verification (e.g., ping-pong lemma on the Bruhat-Tits tree, residual finiteness argument, or exhaustive check of short words in the kernel) is supplied to confirm that the kernel is trivial. This injectivity is load-bearing for both the explicit example and the broader evidence offered for hyperbolic surface groups.

Authors: We agree that merely verifying the relations via substitution establishes a homomorphism but does not by itself prove injectivity. In the revised manuscript we will add an explicit faithfulness argument: we apply a ping-pong lemma on the Bruhat-Tits tree of SL_2 over the function field F_p(x,y), exhibiting two disjoint subsets of the tree on which the generators act with the required contraction/expansion properties. This shows that no non-trivial reduced word in the surface-group generators can lie in the kernel, thereby confirming that the representation is faithful. The argument is self-contained and does not rely on residual finiteness or exhaustive word checks. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit matrices and action evidence are independent of inputs

full rationale

The paper supplies a completely explicit embedding of the genus-2 surface group via concrete matrices in SL_2(F_p(x,y)) and presents evidence for proper actions on tree products. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citation chains appear; the injectivity claim rests on direct substitution into the standard presentation rather than reducing to prior fitted data or author-specific uniqueness theorems. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is supplied; no free parameters, axioms, or invented entities can be extracted from the visible text.

pith-pipeline@v0.9.0 · 5332 in / 1114 out tokens · 27401 ms · 2026-05-15T15:25:25.808202+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

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P-E.Caprace and B.Rémy,Simplicity and superrigidity of twin building lattices, Invent. Math.176(2009) 169—221

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Algebra553(2020) 248–267

M.J.ConderDiscrete and free two-generated subgroups ofSL 2 over non-archimedean local fields, J. Algebra553(2020) 248–267

work page 2020

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GROUPS ACTING ON PRODUCTS OF LOCALLY FINITE TREES 25

G.Conner and M.Mihalik,Commensurated subgroups and ends of groups, J.Group Theory 16(2013) 107—139. GROUPS ACTING ON PRODUCTS OF LOCALLY FINITE TREES 25

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D.Fisher, M.Larsen, R.Spatzier and M.Stover,Character varieties and actions on products of trees, Israel J. Math.225(2018) 889–907

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Cubical-like geometry of quasi-median graphs and applications to geometric group theory

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work page internal anchor Pith review Pith/arXiv arXiv 2017

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work page 2022

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A.LubotzkyandD.Segal, Subgroupgrowth.ProgressinMathematics212, BirkhaüserVerlag, Basel, 2003

work page 2003

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J.-P.Serre, Trees, Springer-Verlag, Heidelberg, 1980

work page 1980

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work page 1979

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D.T.Wise,Complete square complexes, Comment. Math. Helv.82(2007) 683–724. Sel wyn College; University of Cambridge; Cambridge CB3 9DQ; U.K. Email address:j.o.button@cam.ac.uk

work page 2007