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Homological Conditions For Graphical Splittings Of Antisocial Graph Groups

By Michael Batty, James Clerk, Maxwell Building, Mayfield Road and Eh Jz


We define an antisocial graph group to be a graph group arising from a graph whose clique graph is triangle free. In every dimension, the existence of graphical splittings of an antisocial graph group G is shown to correspond to nonvanishing of Betti numbers of the ball of radius 1 in the well-known cube complex on which G acts freely. 1 Introduction. Let # S (G) be the Cayley graph of a finitely generated group G with respect to some finite generating set S and let B n be the ball of radius n in # S (G) about 1 G . If e(n) denotes the number of infinite connected components of # S (G) - B n then the sequence e(n) either converges to 0, 1 or 2 or it diverges to #. We write e(G, S) = lim n## e(n) # Z # {#}. It turns out that e(G, S) is independent of S and we can just write e(G). This is the number of ends of G. In fact it is shown in [B] that if G 1 and G 2 are quasi-isometric groups then e(G 1 ) = e(G 2 ). G is finite if and only if e(G) = 0 and G is virtually cyclic if and ..

Year: 2000
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