Vol. 42, No. 3, 1972

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Cohomology of finitely presented groups

Peter Michael Curran

Vol. 42 (1972), No. 3, 615–620

Let G be a finitely presented group, Ga finite quotient of G and K a field. Let G act on the group algebra V = K[G] in the natural way. For a suitable choice of Gwe obtain estimates on the dimension of H1(G,V ) in terms of the presentation and then use these estimates to derive information about G.

If G is generated by n elements, of which m have finite orders k1,,km, resp., and G has the presentation

⟨a1,⋅⋅⋅ ,an;ak11,⋅⋅⋅ ,akmm,rm+1,⋅⋅⋅ ,rm+q⟩,

then, in particular, we show that (a) the minimum number of generators of G is nq 1∕ki; (b) if this lower bound is actually attained, then G is free, of this rank, and (c) G is infinite if 1∕ki nq 1. The latter, together with a result of R. Fox, yields an algebraic proof that the group

⟨a1,⋅⋅⋅,am;ak11,⋅⋅⋅ , akmm,a1am⟩

is infinite if 1∕ki m 2.

Mathematical Subject Classification 2000
Primary: 20F05
Secondary: 18H10
Received: 27 May 1970
Revised: 21 July 1972
Published: 1 September 1972
Peter Michael Curran