Groups acting on products of locally finite trees

This paper investigates which finitely generated groups admit proper actions on finite products of locally finite trees, presenting evidence that hyperbolic surface groups possess such actions and providing an explicit embedding of the genus 2 surface group into $SL_2(\mathbb{F}_p(x,y))$ for any prime $p$.

The Gist

Imagine you are trying to understand a complex machine (a mathematical group) by seeing how it moves around in a room (a geometric space).

In mathematics, groups are collections of rules or symmetries. To understand them, mathematicians like to see them "acting" on shapes. If a group can move around a simple, curved room (like a hyperbolic space) without getting stuck or crashing into itself, we learn a lot about the group's personality.

This paper, written by J.O. Button, asks a very specific question: Can certain complex groups move around a room made of a finite number of tree-like structures without getting stuck?

Here is a breakdown of the paper's journey, using simple analogies.

1. The Setting: The Room and the Trees

Imagine a "tree" not as a plant, but as a branching network of paths, like a subway map with no loops.

• A Single Tree: If a group tries to move around just one tree, it's very restrictive. Only "virtually free" groups (groups that are basically just free-form wanderers) can do this properly.
• A Product of Trees: Now, imagine a room that is the combination of several trees. Think of it like a 3D grid made of tree branches. If you have two trees, you can move "up/down" on the first tree and "left/right" on the second simultaneously. This creates a much bigger, more flexible playground.

The paper focuses on locally finite trees. This means that at every intersection (vertex) in the tree, there are only a few paths branching off, not an infinite number. It's like a subway station with 3 exits, not a station with a million exits.

2. The Problem: Who Can Play in This Room?

The author is looking for groups that can run around this "tree-room" properly.

• The Easy Players: We know some groups can do this. For example, "Lamplighter groups" (think of a person walking down a street turning lamps on and off) can do it.
• The Impossible Players: Some groups are too "clunky." For instance, if a group contains a specific type of infinite structure (like the Houghton groups), it gets stuck. It's like trying to fit a square peg in a round hole; the group's internal structure forces it to freeze in one spot no matter how it tries to move.

3. The Big Mystery: The Surface Groups

The main character of this story is the Hyperbolic Surface Group.

• The Analogy: Imagine a donut (a torus) or a pretzel with two holes. If you stretch the surface of this shape so it curves negatively (like a Pringles chip), the group of symmetries that keeps this shape intact is the "Surface Group."
• The Question: Can these pretzel-shaped groups run around our "tree-room" properly?
  • We know they can run around a room with infinite trees (where intersections have infinite exits).
  • But can they run around a room with finite trees (where intersections have limited exits)? This has been a hard puzzle for mathematicians.

4. The Detective Work: Finding the Clues

The author doesn't solve the puzzle completely (we don't know for sure yet if they can do it), but he provides very strong evidence that they probably can.

He uses a clever trick involving algebraic fields (which are like number systems with extra rules).

• The Metaphor: Imagine you want to prove a group can run a race. Instead of watching them run, you build a perfect, mathematical "simulator" of the race.
• The author constructs a specific, explicit "simulator" for the genus-2 surface group (the pretzel with two holes). He writes down four specific matrices (grids of numbers) that act as the group's instructions.
• He proves that if you use these instructions in a specific mathematical world (a field of positive characteristic, which is like a clock arithmetic system), the group behaves perfectly. It doesn't crash, and it doesn't get stuck.

5. The "Economical" Solution

The author shows that this group can be embedded into a very specific, small mathematical container: SL(2, Fp(x, y)).

• The Analogy: Think of this as packing a very large, complex suitcase (the group) into a very small, efficient backpack (the field).
• Previous attempts required a huge backpack. The author managed to shrink it down to the smallest possible size for this type of problem.
• Because the group fits so neatly into this small backpack, it suggests that the group is "light" and "agile" enough to run around the tree-room properly.

6. The Conclusion: Strong Evidence, Not a Final Answer

The paper concludes with a "smoking gun" argument:

1. If a group can be represented by these specific numbers in this specific field, it should be able to run around the tree-room.
2. The author built the perfect representation for the surface group.
3. Therefore, it is highly likely that surface groups can act properly on a product of locally finite trees.

In summary:
The paper is like a detective saying, "I haven't caught the suspect (the proof) yet, but I found their fingerprints on the weapon (the explicit matrix representation), and they fit perfectly into the crime scene (the tree product). It's almost certain they did it."

This work bridges the gap between abstract algebra (groups) and geometry (trees), suggesting that the complex, curved world of hyperbolic surfaces is actually compatible with the branching, tree-like world of finite networks.

Technical Summary

Here is a detailed technical summary of the paper "Groups Acting on Products of Locally Finite Trees" by J.O. Button.

1. Problem Statement

The paper investigates the class of finitely generated groups that admit a proper action (where vertex stabilizers are finite) on a finite product of locally finite simplicial trees.

• Context: While geometric group theory often studies groups acting geometrically (properly and cocompactly) on CAT(0) spaces or hyperbolic spaces, such actions require the group to be finitely presented. By relaxing the "cocompact" condition to merely "proper," one can study groups that are finitely generated but not finitely presented (e.g., lamplighter groups).
• The Gap: It is well known that virtually free groups act properly on a single locally finite tree. However, the class of groups acting properly on a product of locally finite trees is less understood.
• The Central Question: Do hyperbolic surface groups ($S_g$, the fundamental group of a closed orientable surface of genus $g \geq 2$) admit a proper action on a finite product of locally finite trees? This is a significant open problem because surface groups are not virtually free, yet they are torsion-free and hyperbolic.

2. Methodology

The author employs a combination of geometric group theory, combinatorial group theory, and representation theory:

1. Geometric Analysis of Actions:

   • The paper analyzes the distinction between actions on arbitrary trees versus locally finite (or bounded valence) trees.
   • It utilizes the concept of cores of tree actions (minimal invariant subtrees) to relate locally finite actions to bounded valence actions for finitely generated groups.
   • It examines fixed point properties (FA, FAlf, FA bv) to identify obstructions.

2. Construction of Counter-Examples and Examples:

   • Lamplighter Groups: The paper constructs explicit proper actions for $F \wr \mathbb{Z}$ on a product of two trees by reversing the standard coset construction (using an ascending HNN extension structure).
   • Houghton Groups: It demonstrates that Houghton groups $H_n$ act properly on a product of $n$ trees (which are not locally finite) but fail to act properly on any product of locally finite trees due to containing "poison" subgroups with fixed point properties.
   • Wreath Products: It proves that $F \wr \mathbb{Z}^2$ acts properly on a product of four trees but not on any product of locally finite trees, using a Newton polygon argument to show that stabilizers become infinite.

3. Representation Theory and Bruhat-Tits Trees:

   • To address the surface group question, the author shifts to linear representations.
   • The strategy relies on the fact that if a group $G$ embeds into $PGL(2, K)$ where $K$ is a global field of positive characteristic, $G$ acts on the associated Bruhat-Tits trees.
   • By selecting a finite set of discrete valuations on $K$, one can induce a proper action on the product of the corresponding trees.
   • The author uses Shalen's Proposition (on amalgamated free products) to construct faithful representations of surface groups (viewed as $F_2 *_{\mathbb{Z}} F_2$) into $SL(2, K)$.

3. Key Contributions and Results

A. Structural Results on Tree Actions

• Equivalence for Finitely Generated Groups: Lemma 2.7 proves that for finitely generated groups, acting properly on a product of locally finite trees is equivalent to acting properly on a product of bounded valence trees.
• Obstructions via Fixed Point Properties: The paper establishes that if a group contains a subgroup $S$ where $S$ and all its finite index subgroups have the fixed point property (FA) (or its locally finite variants), then the group cannot act properly on a product of locally finite trees.
• Theorem 2.9 & Corollary 2.10: The Houghton groups $H_n$ ($n \geq 2$) act properly on a product of $n$ trees but fail to act properly on any product of locally finite trees. This highlights the strictness of the "locally finite" condition.
• Theorem 3.1: The group $F \wr \mathbb{Z}^2$ (where $F$ is finite non-trivial) acts properly on a product of four trees but cannot act properly on any finite product of locally finite trees. This is proven by analyzing the intersection of stabilizers using a Newton polygon argument, showing that the stabilizer of a vertex in the product grows infinitely large.

B. Results on Hyperbolic Surface Groups ($S_g$)

• Necessary Condition (Theorem 4.4): For a torsion-free, finitely generated group $G$ (without $\mathbb{Z} \times \mathbb{Z}$) to act properly on a product of locally finite trees, every non-identity element $g \in G$ must admit a minimal, faithful action on a single locally finite tree where $g$ acts as a loxodromic element.
• Explicit Embeddings (Theorem 5.4 & Corollary 5.5): The paper provides a completely explicit embedding of the genus 2 surface group $S_2$ (and thus all $S_g$ for $g \geq 2$) into $SL(2, \mathbb{F}_p(x, y))$ for any prime $p$.
  • For odd primes, the matrices are defined over $\mathbb{F}_p(x, y)$.
  • For $p=2$, a distinct set of matrices is provided.
  • These embeddings are faithful and satisfy the condition that the commutator of the generators is diagonal.
• Evidence for Proper Action (Corollary 5.6): Based on the explicit embeddings, the author proves that for any finite collection of non-identity elements in $S_g$, there exists a minimal, faithful action on a locally finite tree where all these elements act loxodromically.

4. Significance and Implications

1. Advancement on the Surface Group Problem: While the paper does not definitively prove that $S_g$ acts properly on a product of locally finite trees, it provides the strongest evidence to date. The construction of explicit embeddings into $SL(2, \mathbb{F}_p(x, y))$ (the "most economical" field possible for a 2D representation) suggests that such an action likely exists.
2. Refinement of Obstruction Theory: The paper clarifies the boundary between groups that can and cannot act on products of locally finite trees. It distinguishes between groups that act on products of arbitrary trees (like Houghton groups and $F \wr \mathbb{Z}^2$) and those restricted by the locally finite condition.
3. Methodological Innovation: The use of Newton polygons to analyze the growth of stabilizers in wreath products and the application of valuation theory to construct faithful linear representations of surface groups offer new tools for geometric group theory.
4. Connection to Gromov's Question: The paper notes that if a non-virtually-free hyperbolic group acts properly on a product of locally finite trees, it either contains a surface subgroup (supporting Gromov's conjecture) or provides a counterexample to it. Thus, resolving the surface group case is pivotal for broader conjectures in the field.

In summary, Button's work narrows the search space for groups acting on products of locally finite trees, provides concrete obstructions for specific classes (like $F \wr \mathbb{Z}^2$), and constructs explicit algebraic representations that strongly suggest hyperbolic surface groups belong to this class.