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Let $G$ be a finitely generated group and let $\Gamma=Cay(G, S)$ be the Cayley graph of $G$ with respect to some generating set $S$. If there exists $S$ such that $\Gamma$ is bipartite, then $G$ has …

Yes - see Lemma 5.1 of Gabor Pete's book on probability and groups: http://www.math.bme.hu/~gabor/PGG.html (by the way, these are excellent lecture notes)

Two years ago, Joel Friedman submitted a paper purporting to prove the Hanna Neumann Conjecture, which eventually turned out to contain a fatal bug and was withdrawn. Quite recently, Friedman repeated …

Can anything interesting be deduced about the properties of a group from the behavior of the Ising model on its Cayley graph? (i.e. existence and character of phase transitions, critical behavior) I'm …

Are there any general conditions guaranteeing that the asymptotic cone of a group/graph is "regular" in some sense? E.g. for $\mathbb{Z}^d$ we get $\mathbb{R}^d$ as the asymptotic cone, which is even …

Suppose $G$ is a discrete group and $H \leq G$ a subgroup of finite index. If $H$ has Kazhdan property (T), does it follow that $G$ has property (T)? (I've read somewhere that (T) is preserved by exac …

Let $G$ be a finite group. Do eigenvalues of its Cayley graph say anything about the algebraic properties of $G$? The spectrum of Cayley graph may depend on the presentation, so it's not a good invari …

There is a very nice book related to the topic - "Word processing in groups" by David Epstein. It covers some stuff about the combinatorial aspects of geometric group theory, e.g. automatic groups, co …