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H. Bass has studied existence of lattices on trees. Can someone suggest a (readable) reference for lattices on graphs?

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    Yes, of course, Bass's existence theorem is one of the deep results on tree lattices. I tried to point out to you that you can't expect to say anything sensible about lattices in groups of automorphisms of (locally finite) graphs without further restrictions (e.g. Gromov hyperbolicity) since otherwise you'll get *all* finitely generated groups.2011-08-31

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