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I have seen that an important tool of finitely presented groups consists in writing down its Cayley graph with respect to a given set of generators, and then try to extract data like the coarse geometry or other things...

Does there exist a related theory for finitely generated Lie algebra? I guess one of the first difficulties here is that a word [a,[b,c]] would decompose into a linear combination of words...

Thank you for any answer.

Klaus

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  • $\begingroup$ Cayley graphs are defined in the same way for finitely generated groups, the data is just one group and one finite generating family. No presentation is involved. $\endgroup$
    – YCor
    Mar 6, 2015 at 18:25
  • $\begingroup$ I am confused. Are you claiming that there are Cayley graphs for finitely generated Lie algebras? I also don't understand why you say that the presentation is not interesting for the Cayley graph of a group? As far as I understand the Cayley graph of a group has as vertices the elements of the group. Then you connect to vertices $g$ and $h$ if $g = sh$ for some $s$ in the generating set $S$. But I guess that to decide whether "$g=sh$" you do need to use the relations. I don't work in algebra, so I'm sorry if I'm just mixing up things... $\endgroup$ Mar 7, 2015 at 12:02
  • $\begingroup$ No. I said that Cayley graphs for groups are defined regardless of any presentation. I did't say anything about Lie algebras. But if there's an analogy, it should be with finitely generated Lie algebras. $\endgroup$
    – YCor
    Mar 7, 2015 at 12:04

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