Suppose I have a finitely generated group $G$ acting on a set $X,$ and we make a graph, whose vertex set is $X,$ and where we join $x_1$ to $x_2$ by an edge labeled $g_i$ if $g_i(x_1) = x_2.$ What is this called? Is it still a Cayley graph?
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asked Oct 20, 2016 at 14:10
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2$\begingroup$ It's the union of complete graphs on orbits... more generally when you only do it for $g$ in a given generating set, it's called Schreier graph (not Cayley graph). $\endgroup$– YCorCommented Oct 20, 2016 at 14:16
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$\begingroup$ (Actually this is a labeled graph allowing multiplicities, so in the first case it's a bit more that just the complete graph... it called the action groupoid.) $\endgroup$– YCorCommented Oct 20, 2016 at 14:35
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$\begingroup$ @YCor $g_i$ is a generator, not an arbitrary element, so I guess Schreier graph is it... $\endgroup$– Igor RivinCommented Oct 20, 2016 at 14:58
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2$\begingroup$ It is actually a category, not (only) a graph. It is called action groupoid. $\endgroup$– Gejza JenčaCommented Oct 20, 2016 at 19:25
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