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We use elementary methods to compute the L 2-dimension of the eigen-spaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamp-lighter group found by Grigorchuk-Zuk [4]. The latter result was used(More)
We determine the L 2-Betti numbers of all one-relator groups and all surface plus one relation groups. We also obtain some information about the L 2-cohomology of left-orderable groups, and deduce the non-L 2 result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free. 1 Notation and background Let G be a(More)
For each finite ordinal n, and each locally-finite group G of cardinality ℵ n , we construct an (n + 1)-dimensional, contractible CW-complex on which G acts with finite stabilizers. We use the complex to obtain information about cohomology with induced coefficients. Our techniques also give information about the location of some large free abelian groups in(More)
Submultiplicativity, an analytic property generalizing the Strengthened Hanna Neumann Conjecture (SHNC) to complexes was proved in [2] assuming the deep-fall property. This in particular implied SHNC. The purpose of this note is to write the proof of the original SHNC and purely in terms of groups and graphs. We also give explicit examples showing that the(More)
Let R be a ring, let F be a free group, and let X be a basis of F. Let : RF → R denote the usual augmentation map for the group ring RF , let X∂ := {x − 1 | x ∈ X} ⊆ RF , let Σ denote the set of matrices over RF that are sent to invertible matrices by , and let (RF)Σ −1 denote the universal localization of RF at Σ. A classic result of Magnus and Fox gives(More)
To Martin Dunwoody on the occasion of his 70th birthday. Abstract This article surveys many standard results about the braid group, with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the braid group considered as the algebraic mapping-class group of a(More)