---
tag: 4XQ8TYS2C4
kind: definition
name: Cayley–Abels graph
msc: [22D05, 20F65, 05C25]
defines: QWG66JE1JE
uses: [AHV05ZBCYW, AW3J9AQSA0, QYCFH68RD1, 0KWFFMDF8T, 0KS0KRDV4Y, 02GK6HZYBT, 0GYQK9NDJF]
sources:
  - url: https://en.wikipedia.org/wiki/Cayley_graph
---

## Definition

Let $G$ be a [[totally disconnected|AW3J9AQSA0]] [[locally compact group|AHV05ZBCYW]] that is compactly generated.
A *Cayley–Abels graph* for $G$ is a graph $\Gamma$ with an action of $G$ by graph automorphisms such that:

1. $\Gamma$ is [[connected|0KWFFMDF8T]];
2. $\Gamma$ is locally finite: every vertex has finite [[degree|0KS0KRDV4Y]];
3. $G$ acts [[transitively|02GK6HZYBT]] on the vertices of $\Gamma$;
4. the stabiliser in $G$ of each vertex is a compact [[open subgroup|QYCFH68RD1]] of $G$.

## Remarks

Example: let $G$ be a finitely generated group with the discrete topology, and let $S$ be a finite symmetric generating set.
The [[Cayley graph|0GYQK9NDJF]] of $(G, S)$, with $G$ acting by left multiplication, is connected because $S$ generates $G$.
Each vertex has degree at most $\abs{S}$, the action is transitive, and stabilisers are trivial, hence compact and open.

Example: $G = \mathrm{Aut}(T_d)$, the automorphism group of the regular tree $T_d$ of degree $d \ge 3$, acting on $T_d$.
Vertex stabilisers are compact and open in the topology of pointwise convergence.

Non-example: $\Z$ acting on its Cayley graph with respect to the infinite generating set $\Z$.
The graph is complete, so every vertex has infinite degree, and clause 2 fails.

Construction: choose a compact open subgroup $U$ and a finite symmetric set $A$ with $G$ generated by $U \cup A$.
Take the cosets $gU$ as vertices, and join $gU$ and $hU$ when $g^{-1} h \in U A U$.
This is a Cayley–Abels graph, since $UAU$ is a finite union of cosets of $U$.