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  "body": "\n## Definition\n\nLet $G$ be a [[totally disconnected|AW3J9AQSA0]] [[locally compact group|AHV05ZBCYW]] that is compactly generated.\nA *Cayley–Abels graph* for $G$ is a graph $\\Gamma$ with an action of $G$ by graph automorphisms such that:\n\n1. $\\Gamma$ is [[connected|0KWFFMDF8T]];\n2. $\\Gamma$ is locally finite: every vertex has finite [[degree|0KS0KRDV4Y]];\n3. $G$ acts [[transitively|02GK6HZYBT]] on the vertices of $\\Gamma$;\n4. the stabiliser in $G$ of each vertex is a compact [[open subgroup|QYCFH68RD1]] of $G$.\n\n## Remarks\n\nExample: let $G$ be a finitely generated group with the discrete topology, and let $S$ be a finite symmetric generating set.\nThe [[Cayley graph|0GYQK9NDJF]] of $(G, S)$, with $G$ acting by left multiplication, is connected because $S$ generates $G$.\nEach vertex has degree at most $\\abs{S}$, the action is transitive, and stabilisers are trivial, hence compact and open.\n\nExample: $G = \\mathrm{Aut}(T_d)$, the automorphism group of the regular tree $T_d$ of degree $d \\ge 3$, acting on $T_d$.\nVertex stabilisers are compact and open in the topology of pointwise convergence.\n\nNon-example: $\\Z$ acting on its Cayley graph with respect to the infinite generating set $\\Z$.\nThe graph is complete, so every vertex has infinite degree, and clause 2 fails.\n\nConstruction: choose a compact open subgroup $U$ and a finite symmetric set $A$ with $G$ generated by $U \\cup A$.\nTake the cosets $gU$ as vertices, and join $gU$ and $hU$ when $g^{-1} h \\in U A U$.\nThis is a Cayley–Abels graph, since $UAU$ is a finite union of cosets of $U$.\n",
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