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A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this situation. So we need to look for infinitely generating groups to find a Cayley graph with infinite discrete end space.

Any further references on Cayley graphs of infinitely generated groups would be appreciated, I can't seem to find much on this.

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  • $\begingroup$ The answer is in your question, no? Or if this is a reference request, what do you want exactly? $\endgroup$
    – Ville Salo
    Commented Dec 2 at 18:16
  • $\begingroup$ @Ville I guess the question is about Cayley graphs w.r.t. infinite generating sets. $\endgroup$
    – Corentin B
    Commented Dec 2 at 18:24
  • $\begingroup$ Yes, that's what I meant, that we would have to look at infinite generating sets for an answer. Let me edit to clarify. $\endgroup$
    – violeta
    Commented Dec 2 at 19:04
  • $\begingroup$ Could you be more specific on how you define the space of ends for a general connected graph? $\endgroup$
    – YCor
    Commented Dec 3 at 9:28
  • $\begingroup$ I'm considering the definition of ends used in infinite graph theory: you define a ray as an infinite path, and an end is an equivalence class of rays, where the equivalence is given by saying two rays are equivalent when they are infinitely connected (as in, removing any finite amount of vertices does not disconnect them). Here's a reference: en.wikipedia.org/wiki/End_(graph_theory) $\endgroup$
    – violeta
    Commented Dec 3 at 14:02

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