Groups acting on trees with Tits' independence property (P)

Colin D. Reid; Simon M. Smith

arxiv: 2002.11766 · v3 · submitted 2020-02-26 · 🧮 math.GR

Groups acting on trees with Tits' independence property (P)

Colin D. Reid , Simon M. Smith This is my paper

Pith reviewed 2026-05-24 15:15 UTC · model grok-4.3

classification 🧮 math.GR

keywords Tits independence propertygroup actions on treeslocal actionsBass-Serre theorylocally compact groupssimple groupsgeometric density

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The pith

Closed group actions on trees with Tits independence property (P) decompose into and reconstruct from local action diagrams.

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

The paper develops a coherent theory of local actions that classifies every closed group action on a tree satisfying Tits' independence property (P). Any such action decomposes into a local action diagram, a decorated graph that records exactly how each vertex stabiliser acts on the neighbouring vertices. From any valid diagram one can construct a corresponding closed group action on a tree. Properties such as geometric density, compact generation and simplicity are then readable directly from the diagram. The resulting framework complements classical Bass-Serre theory and organises the local-to-global passage used in constructions of nondiscrete simple groups.

Core claim

In this article we give a full classification and description of all closed group actions on trees with Tits' independence property (P) using a new coherent theory for local actions that applies to all actions on trees. This theory is a local action complement to classical Bass-Serre theory. On the one hand, our theory gives a decomposition of a group acting on a tree into a local action diagram, a decorated graph that encodes all local information, and on the other hand a construction of a group acting on a tree from a given local action diagram. One can read directly from the local action diagram whether the resulting group has certain properties, like geometric density, compact generation

What carries the argument

The local action diagram: a decorated graph encoding the local actions of vertex stabilisers on their neighbours, serving as the complete invariant under property (P).

If this is right

Geometric density of the group is determined by inspecting the local action diagram.
Compact generation of the group is determined by inspecting the local action diagram.
Simplicity of the group is determined by inspecting the local action diagram.
Every closed action with property (P) arises from some local action diagram.

Where Pith is reading between the lines

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

The diagrams provide a uniform way to generate all previously known examples of nondiscrete simple groups built from property (P).
Converting an existing action to its diagram allows immediate verification of its listed properties without separate arguments.
The same diagram language may organise actions that preserve additional structure such as a measure or a metric.

Load-bearing premise

A single coherent local-action theory exists that applies uniformly to all closed actions on trees with property (P) and yields both a complete decomposition into diagrams and a faithful reconstruction from any diagram.

What would settle it

An explicit closed group action on a tree with independence property (P) that cannot be expressed as a local action diagram, or a local action diagram whose reconstruction fails to satisfy property (P).

read the original abstract

Local actions (actions of a vertex stabiliser on the neighbours of that vertex) have become an important approach to group actions on trees since J. Tits' introduction in 1970 of the independence property (P) and especially since a 2000 paper by M. Burger and Sh. Mozes. This `local-to-global' approach has been critical in the development of the theory of totally disconnected locally compact groups because it allows the construction of nondiscrete group actions on trees while keeping control over the action of a vertex stabiliser, in a way that is not practical under the classical Bass-Serre approach. The majority of constructions of nonlinear nondiscrete locally compact simple groups use (P) and its generalisations. In this article we give a full classification and description of all closed group actions on trees with Tits' independence property (P) using a new coherent theory for local actions that applies to all actions on trees. This theory is a `local action' complement to classical Bass-Serre theory. On the one hand, our theory gives a decomposition of a group acting on a tree into a `local action diagram' (a decorated graph that encodes all `local' information), and on the other hand a construction of a group acting on a tree from a given local action diagram. One can read directly from the local action diagram whether the resulting group has certain properties, like geometric density, compact generation and simplicity.

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

Reid and Smith give a local-action diagram that decomposes closed tree actions with (P) and reconstructs the group from the diagram with properties like simplicity readable off it, but the gluing step may require unstated compatibility conditions.

read the letter

The main point is that this paper sets up a local action diagram as a decorated graph encoding the vertex stabilizers' actions on neighbors, then claims both a decomposition of any closed G ≤ Aut(T) with (P) into such a diagram and a faithful construction back from any diagram. Properties such as geometric density, compact generation, and simplicity are supposed to be readable directly from the diagram. This is positioned as a local-to-global complement to Bass-Serre theory, aimed at the constructions of nondiscrete tdlc simple groups that have relied on (P) since Tits and Burger-Mozes.

Referee Report

2 major / 1 minor

Summary. The paper develops a 'local action' theory as a complement to Bass-Serre theory for closed subgroups G ≤ Aut(T) satisfying Tits' independence property (P). It claims a decomposition of any such G into a decorated graph called a local action diagram that encodes all local permutation-group data at vertices, together with a faithful inverse construction that produces a closed group with (P) from any such diagram; properties such as geometric density, compact generation and simplicity are asserted to be readable directly from the diagram.

Significance. If the decomposition and reconstruction are fully faithful and preserve (P) and closedness without extra conditions, the work would supply a uniform, diagram-based classification that strengthens the local-to-global methods introduced by Tits and developed by Burger-Mozes, with direct applications to the construction of nondiscrete simple td lc groups.

major comments (2)

[reconstruction theorem] The reconstruction theorem (presumably §3 or §4) asserts that an arbitrary local action diagram can be glued to produce a closed subgroup with (P). The manuscript must explicitly verify that the edge-stabilizer compatibility required for (P) at adjacent vertices is automatically satisfied by the diagram data alone; otherwise the inverse construction is not faithful for all inputs and the classification is incomplete.
[decomposition theorem] The decomposition theorem claims that every closed G with (P) arises from a unique local action diagram. The proof must show that the diagram is independent of the choice of fundamental domain or generating set; if the extraction of local actions at vertices requires additional global closedness arguments that are not functorial, the correspondence fails to be bijective.

minor comments (1)

Notation for the decorated graph (vertices labelled by local permutation groups, edges by stabilizers) should be introduced with a single running example that is carried through both the decomposition and reconstruction sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on the reconstruction and decomposition theorems. We agree that greater explicitness on certain verifications would improve the manuscript and will incorporate clarifications in a revised version.

read point-by-point responses

Referee: [reconstruction theorem] The reconstruction theorem (presumably §3 or §4) asserts that an arbitrary local action diagram can be glued to produce a closed subgroup with (P). The manuscript must explicitly verify that the edge-stabilizer compatibility required for (P) at adjacent vertices is automatically satisfied by the diagram data alone; otherwise the inverse construction is not faithful for all inputs and the classification is incomplete.

Authors: We thank the referee for this observation. The local action diagram is defined so that it includes the full data of the edge stabilizers together with their embeddings into the adjacent vertex actions; the reconstruction theorem then assembles the group by taking the appropriate inverse limit over these data. The proof already verifies that this assembly automatically satisfies the edge-stabilizer compatibility needed for (P), because the diagram encodes precisely the required inclusions and actions. To make the argument more transparent, we will insert a short dedicated paragraph (or lemma) immediately after the statement of the reconstruction theorem that isolates this compatibility check. revision: partial
Referee: [decomposition theorem] The decomposition theorem claims that every closed G with (P) arises from a unique local action diagram. The proof must show that the diagram is independent of the choice of fundamental domain or generating set; if the extraction of local actions at vertices requires additional global closedness arguments that are not functorial, the correspondence fails to be bijective.

Authors: The decomposition extracts the local action diagram directly from the conjugacy classes of vertex and edge stabilizers together with their natural actions on the link; because the group is closed and satisfies (P), these data are intrinsic and do not depend on a particular fundamental domain or generating set. The uniqueness part of the proof shows that any two diagrams obtained from different choices are canonically isomorphic. We will revise the decomposition section to add an explicit remark confirming independence from the choice of domain and to note the functoriality of the extraction map. revision: partial

Circularity Check

0 steps flagged

No circularity: new local-action theory is constructed from external Tits/Burger-Mozes foundations

full rationale

The paper introduces a decomposition into local action diagrams and a reconstruction construction as a complement to Bass-Serre theory. It explicitly grounds the independence property (P) in Tits (1970) and Burger-Mozes (2000), with the classification presented as a uniform mathematical framework applicable to all closed actions with (P). No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or claimed derivation chain. The gluing and faithfulness claims are part of the new theory's content rather than reductions to prior inputs by construction. This is the normal case of an independent development.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of a uniform local-action theory that works for every action on trees; the independence property (P) is treated as given domain data rather than derived.

axioms (1)

domain assumption Tits' independence property (P) holds for the local actions under consideration
The classification applies precisely to actions satisfying this property, as stated in the abstract.

invented entities (1)

local action diagram no independent evidence
purpose: A decorated graph encoding all local information that permits both decomposition of a given action and reconstruction of the group from the diagram
Introduced as the central new object of the theory; no independent evidence outside the paper is supplied in the abstract.

pith-pipeline@v0.9.0 · 5789 in / 1286 out tokens · 27209 ms · 2026-05-24T15:15:20.731506+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2. There is a natural one-to-one correspondence between conjugacy classes of (P)-closed actions on trees and isomorphism classes of local action diagrams.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.1. A local action diagram Δ = (Γ, (Xa), (G(v))) … G(v) is a closed subgroup of Sym(Xv) …

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Banks, M

C. Banks, M. Elder and G. A. Willis, Simple groups of autom orphisms of trees determined by their actions on ﬁnite subtrees. J. Group Theory 18 (2015), 235–261

work page 2015
[2]

Burger and Sh

M. Burger and Sh. Mozes, Groups acting on trees: from loca l to global structure. Publ. Math. IH ´ES 92 (2000), 113–150

work page 2000
[3]

Caprace and P

P.-E. Caprace and P. R. Wesolek, Indicability, residual ﬁniteness, and simple subquotients of groups acting on trees. Geom. Topol. 22 (2018), no. 7, 4163–4204

work page 2018
[4]

Castellano, Rational discrete ﬁrst degree cohomolog y for totally dis- connected locally compact groups

I. Castellano, Rational discrete ﬁrst degree cohomolog y for totally dis- connected locally compact groups. Math. Proc. Camb. Philos. Soc. 168 (2020), no. 2, 361–377

work page 2020
[5]

The GAP Group, GAP – Groups, Algorithms, and Programming , Ver- sion 4.8.7 (2017), http://www.gap-system.org

work page 2017
[6]

Holt, Enumerating subgroups of the symmetric group

D. Holt, Enumerating subgroups of the symmetric group. I n: Compu- tational Group Theory and the Theory of Groups, II , edited by L.-C. Kappe, A. Magidin and R. Morse (2010), AMS Contemporary Math e- matics book series, vol. 511, pp. 33–37

work page 2010
[7]

R. G. M¨ oller, FC ´ -elements in totally disconnected groups and auto- morphisms of inﬁnite graphs. Math. Scand. 92 (2003), 261–268

work page 2003
[8]

R. G. M¨ oller and J. Vonk, Normal subgroups of groups acti ng on trees and automorphism groups of graphs. J. Group Theory 15 (2012), no. 6, 831–850

work page 2012
[9]

J. P. Serre, Trees. Springer Monogr. in Math., Springer, Berlin, 2003

work page 2003
[10]

Sloane, The On-Line Encyclopedia of Integer Seq uences, pub- lished electronically at https://oeis.org

N.J.A. Sloane, The On-Line Encyclopedia of Integer Seq uences, pub- lished electronically at https://oeis.org

work page
[11]

S. M. Smith, A product for permutation groups and topolo gical groups. Duke Math. J. 166 (2017), no. 15, 2965–2999

work page 2017
[12]

Tits, Sur le groupe des automorphismes d’un arbre

J. Tits, Sur le groupe des automorphismes d’un arbre. In : Essays on topology and related topics (M´ emoires d´ edi´ es ` a Georges de Rham) , Springer, New York, 1970, pp. 188–211. 56

work page 1970

[1] [1]

Banks, M

C. Banks, M. Elder and G. A. Willis, Simple groups of autom orphisms of trees determined by their actions on ﬁnite subtrees. J. Group Theory 18 (2015), 235–261

work page 2015

[2] [2]

Burger and Sh

M. Burger and Sh. Mozes, Groups acting on trees: from loca l to global structure. Publ. Math. IH ´ES 92 (2000), 113–150

work page 2000

[3] [3]

Caprace and P

P.-E. Caprace and P. R. Wesolek, Indicability, residual ﬁniteness, and simple subquotients of groups acting on trees. Geom. Topol. 22 (2018), no. 7, 4163–4204

work page 2018

[4] [4]

Castellano, Rational discrete ﬁrst degree cohomolog y for totally dis- connected locally compact groups

I. Castellano, Rational discrete ﬁrst degree cohomolog y for totally dis- connected locally compact groups. Math. Proc. Camb. Philos. Soc. 168 (2020), no. 2, 361–377

work page 2020

[5] [5]

The GAP Group, GAP – Groups, Algorithms, and Programming , Ver- sion 4.8.7 (2017), http://www.gap-system.org

work page 2017

[6] [6]

Holt, Enumerating subgroups of the symmetric group

D. Holt, Enumerating subgroups of the symmetric group. I n: Compu- tational Group Theory and the Theory of Groups, II , edited by L.-C. Kappe, A. Magidin and R. Morse (2010), AMS Contemporary Math e- matics book series, vol. 511, pp. 33–37

work page 2010

[7] [7]

R. G. M¨ oller, FC ´ -elements in totally disconnected groups and auto- morphisms of inﬁnite graphs. Math. Scand. 92 (2003), 261–268

work page 2003

[8] [8]

R. G. M¨ oller and J. Vonk, Normal subgroups of groups acti ng on trees and automorphism groups of graphs. J. Group Theory 15 (2012), no. 6, 831–850

work page 2012

[9] [9]

J. P. Serre, Trees. Springer Monogr. in Math., Springer, Berlin, 2003

work page 2003

[10] [10]

Sloane, The On-Line Encyclopedia of Integer Seq uences, pub- lished electronically at https://oeis.org

N.J.A. Sloane, The On-Line Encyclopedia of Integer Seq uences, pub- lished electronically at https://oeis.org

work page

[11] [11]

S. M. Smith, A product for permutation groups and topolo gical groups. Duke Math. J. 166 (2017), no. 15, 2965–2999

work page 2017

[12] [12]

Tits, Sur le groupe des automorphismes d’un arbre

J. Tits, Sur le groupe des automorphismes d’un arbre. In : Essays on topology and related topics (M´ emoires d´ edi´ es ` a Georges de Rham) , Springer, New York, 1970, pp. 188–211. 56

work page 1970