Abstract
A one-relator group is a group G(r) that admits a presentation < S vertical bar r > with a single relation r. One-relator groups form a rich classically studied class of groups in geometric group theory. If r is an element of F(S)', we introduce a simplicial volume kGrk for one-relator groups. We relate this invariant to the stable commutator length scl(S)(r) of the element r is an element of ...
Abstract
A one-relator group is a group G(r) that admits a presentation < S vertical bar r > with a single relation r. One-relator groups form a rich classically studied class of groups in geometric group theory. If r is an element of F(S)', we introduce a simplicial volume kGrk for one-relator groups. We relate this invariant to the stable commutator length scl(S)(r) of the element r is an element of F(S)'. We show that often (though not always) the linear relationship parallel to G(r)parallel to D4 scl(S)(r) - 2 holds and that every rational number modulo 1 is the simplicial volume of a one-relator group. Moreover, we show that this relationship holds approximately for proper powers and for elements satisfying the small cancellation condition C'(1,N), with a multiplicative error of O(1, N). This allows us to prove for random elements of F.S/0 of length n that parallel to G(r)parallel to is 2/3 log(2 vertical bar S vertical bar 1). n=log nCo(n=log n) with high probability, using an analogous result of Calegari and Walker for stable commutator length.
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