PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-26

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Cayley-graph neighbourhoods are equipotent, and local finiteness is equivalent to finiteness of the symmetrised subset

Statement

Let $G$ be a group, $S \subseteq G$ , and $S^{\pm} : = (S \cup S^{- 1}) ∖ {e}$ . Left translation gives a bijection between the neighbourhoods of any two vertices of $Cay (G, S)$ . The graph is locally finite exactly when $S^{\pm}$ is finite; in that case it is regular of finite degree $∣ S^{\pm} ∣$ .

Facts & Assumptions

Given: A group $G$ , a subset $S \subseteq G$ , and $S^{\pm} = (S \cup S^{- 1}) ∖ {e}$ .

[F1]

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set ${{g, g s} : g \in G, s \in (S \cup S^{- 1}) ∖ {e}}$ (The Cayley graph of a group with respect to a subset).

[L1]

A graph is locally finite when every vertex has finitely many neighbours (Locally finite graphs and vertex degree without a finiteness hypothesis).

[L2]

The degree of $v$ is $\deg_{G} (v) : = ∣ N_{G} (v) ∣$ , equivalently the number of edges incident with $v$ . A graph is $r$ -regular when every vertex has degree $r$ ; it is cubic when it is $3$ -regular. (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[L3]

A set $A$ is finite when $A \approx n$ for some $n \in N$ . (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct

1.1F1L2

The neighbours of $g$ are the elements $g s$ with $s$ in the symmetrised set minus the identity, and left multiplication by $h g^{- 1}$ is a bijection from the neighbours of $g$ to those of $h$ .

2.1F1L1L2L3step 1.1∎

Thus one neighbourhood is finite exactly when all are, which occurs exactly when $S^{\pm}$ is finite. In that case the degree is defined at every vertex and equals $∣ S^{\pm} ∣$ , so the graph is regular of that finite degree.

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)