= Cayley graph
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{wiki}
A <Cayley graph> turns multiplication in a <group> into travel: the chosen <generators of a group> are the allowed one-step moves, and a <word over an alphabet> becomes a route. This lets us study a <finitely generated group> using <graph distances>, <closed balls>, and <geodesics>, rather than only equations between products.
Let $G$ be a <group> with a finite <generating set of a group> $S$ satisfying $S=S^{-1}$ and $1\notin S$. Such a <symmetric subset of a group> includes the <inverse element> of every allowed move. The <Cayley graph> $\operatorname{Cay}_S(G)$ has <vertex set> $G$ and joins $g$ to $gs$ for each $s\in S$. We choose right multiplication for these moves. Reversing the move uses $s^{-1}$, so for <graph distance> we regard $g\to gs$ and $gs\to g$ as the two orientations of one undirected <edge>. An involution therefore gives one <edge>, not two parallel <edges>. Each <edge> has length one, regardless of its length in a drawing.https://people.maths.ox.ac.uk/bridson/papers/BATH97.pdf[Bridson, Non-positive curvature in group theory, Definition 1.7]{ref}
A sequence $s_1,\ldots,s_n$ labels the <walk>
$$
g,\quad gs_1,\quad gs_1s_2,\quad\ldots,\quad gs_1\cdots s_n.
$$
It ends at $h$ exactly when $g^{-1}h=s_1\cdots s_n$. Thus the <Cayley graph> is a <connected graph> precisely because $S$ generates $G$, and its <graph distance> on <vertices> is the <word metric>. Left multiplication by $a$ sends the <edge> from $g$ to $gs$ to the <edge> from $ag$ to $(ag)s$, preserving its generator label: this is a <group action> by <graph automorphisms> and <isometries>. Viewing <groups> as symmetries of geometric objects is the motivation of Clay and Margalit's introduction.https://assets.press.princeton.edu/chapters/s11042.pdf[Clay and Margalit, Office Hours with a Geometric Group Theorist, Office Hour 1, pages 3–5]{ref}
The useful geometric object includes its allowed moves, not just its drawing. <Cayley graphs of the integers> show how changing the moves adds shortcuts; <word metric balls in the integer lattice> show how the same <group> can acquire visibly different <closed balls>. Meier's book develops this passage from <group operations> to <graphs> as its central motivation.https://api.pageplace.de/preview/DT0400.9780511421594_A23678234/preview-9780511421594_A23678234.pdf[Meier, Groups, Graphs and Trees, Preface and Chapter 1 introduction]{ref}
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