Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

2006/12/14 by Goulnara Arzhantseva, Arzhantseva, Goulnara, Cornelia Druţu +3
Mathematics · #20E06 #20F65 #20F69 #46B07 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.math/0612378

openalex publication_date 2006/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct finitely generated groups with arbitrary prescribed Hilbert space compression αfrom 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.

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