265 research outputs found
Compression functions of uniform embeddings of groups into Hilbert and Banach spaces
We construct finitely generated groups with arbitrary prescribed Hilbert
space compression \alpha from the interval [0,1]. For a large class of Banach
spaces E (including all uniformly convex Banach spaces), the E-compression of
these groups coincides with their Hilbert space compression. Moreover, the
groups that we construct have asymptotic dimension at most 3, hence they are
exact. In particular, the first examples of groups that are uniformly
embeddable into a Hilbert space (respectively, exact, of finite asymptotic
dimension) with Hilbert space compression 0 are given. These groups are also
the first examples of groups with uniformly convex Banach space compression 0.Comment: 21 pages; version 3: The final version, accepted by Crelle; version
2: corrected misprints, added references, the group has asdim at most 2, not
at most 3 as in the first version (thanks to A. Dranishnikov); version 3:
took into account referee remarks, added references. the paper is accepted in
Crell
Compression functions of uniform embeddings of groups into Hilbert and Banach spaces
We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. This answers a question of E. Guentner and G. Niblo. For a large class of Banach spaces ℰ (including all uniformly convex Banach spaces), the ℰ-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 2, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (moreover, of finite asymptotic dimension and exact) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression
Non-coherence of arithmetic hyperbolic lattices
We prove, under the assumption of the virtual fibration conjecture for
arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n>
4, and different from 7, are non-coherent. We also establish noncoherence of
uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of
uniform lattices in SU(2,1) which have infinite abelianization.Comment: 26 pages, 3 figure
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