A. Dranishnikov, J. Smith (2006)
Fundamenta Mathematicae
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We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length.
A. Ivanov (1999)
Colloquium Mathematicae
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We study infinite finitely generated groups having a finite set of conjugacy classes meeting all cyclic subgroups. The results concern growth and the ascending chain condition for such groups.
Dmitry Matsnev (2008)
Colloquium Mathematicae
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We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov, and we give an improved estimate of that dimension in terms of the relator length. The construction is similar to one of Bell and Dranishnikov, but we produce a sharper estimate.
Otera, Daniele Ettore (2008)
Acta Universitatis Apulensis. Mathematics - Informatics
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Andrzej Strojnowski (1999)
Colloquium Mathematicae
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The paper is concerned with the class of groups satisfying the finite embedding (FE) property. This is a generalization of residually finite groups. In [2] it was asked whether there exist FE-groups which are not residually finite. Here we present such examples. To do this, we construct a family of three-generator soluble FE-groups with torsion-free abelian factors. We study necessary and sufficient conditions for groups from this class to be residually finite. This answers the questions...
Yves Cornulier (2011)
Annales de l’institut Fourier
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We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank .
G. C. Bell, A. N. Dranishnikov, J. E. Keesling (2004)
Fundamenta Mathematicae
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We prove an exact formula for the asymptotic dimension (asdim) of a free product. Our main theorem states that if A and B are finitely generated groups with asdim A = n and asdim B ≤ n, then asdim (A*B) = max n,1.
Yang Kok Kim (1996)
Rendiconti del Seminario Matematico della Università di Padova
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Bartholdi, Laurent, Grigorchuk, Rostislav (2002)
Serdica Mathematical Journal
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* The authors thank the “Swiss National Science Foundation” for its support. We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations....
Romain Tessera (2006-2007)
Séminaire de théorie spectrale et géométrie
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We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the...
Christophe Pittet (1995)
Revista Matemática Iberoamericana
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The isoperimetric inequality |∂Ω| / |Ω| = constant / log |Ω| for finite subsets Ω in a finitely generated group Γ with exponential growth is optimal if Γ is polycyclic.