Relative subgroup growth and subgroup distortion

Davis, Tara C.; Olshanskii, A. Yu.

doi:10.4171/ggd/312

Cited by 9 publications

(9 citation statements)

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“…If K and N have subrecursive Følner functions and the distortion function ∆ Γ N (n) := max{|ω| Y : ω ∈ N, |ω| X ≤ n} is subrecursive, then Γ has subrecursive Følner function; if N has computable Følner sets, K is amenable with solvable word problem and ∆ Γ N is subrecursive then Γ has computable Følner sets. Notice that it is possible that ∆ Γ N is not subrecursive, see for example [1], even for solvable groups, see [6]. Again we don't know if these hypotheses are necessary.…”

Section: Theorem a The Kharlampovich Groups G(m) Have Computable Følmentioning

confidence: 99%

Computability of Følner sets

Cavaleri

2017

Int. J. Algebra Comput.

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ABSTRACT. We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions -with subrecursive distortion functions-of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets . Moreover we obtain some known and some new upper bounds for the Følner function for these particular extensions.

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Section: Theorem a The Kharlampovich Groups G(m) Have Computable Følmentioning

confidence: 99%

Computability of Følner sets

Cavaleri

2017

Int. J. Algebra Comput.

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“…The geodesic distance in a wreath product has been especially studied for Cayley graphs by the investigation of the Word length for wreath products of finite and infinite groups [7,10,23]. In any approach it appears an NP-hard problem: the Traveling Salesman Problem (TSP).…”

Section: Distances In a Wreath Productmentioning

confidence: 99%

“…Remark 4.12. In the works [7,10] an analog formula is given for the word length of an element in a wreath product of finitely generated groups, via canonical form.…”

Section: Distances In a Wreath Productmentioning

confidence: 99%

Some degree and distance-based invariants of wreath products of graphs

Cavaleri

Donno

2020

Discrete Applied Mathematics

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The wreath product of graphs is a graph composition inspired by the notion of wreath product of groups, with interesting connections with Geometric Group Theory and Probability. This paper is devoted to the description of some degree and distancebased invariants, of large interest in Chemical Graph Theory, for a wreath product of graphs. An explicit formula is obtained for the Zagreb indices, in terms of the Zagreb indices of the factor graphs. A detailed analysis of distances in a wreath product is performed, allowing to describe the antipodal graph and to provide a formula for the Wiener index. Finally, a formula for the Szeged index is obtained. Several explicit examples are given. (2010): 05C07, 05C12, 05C40, 05C76. Mathematics Subject Classification

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“…Further consider the unrestricted wreath product V Wr Z, where V is defined by (8). Let U be the union of sets U 1 and U 2 defined as follows: U 1 consists of all functions f 1,h from the base of V Wr Z such that…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…In the corollary below we give a complete description of ordinals that can be realized as elementary classes of countable elementary amenable groups. Yet another application of our main result will be given in the forthcoming joint paper [8] by Tara Davis and the first author: Every super-additive function N ∪ {0} → N ∪ {0} can be realized (up to equivalence) as the distortion function of a cyclic subgroup in a finitely generated solvable group. (For definitions and background see Gromov's paper [12]).…”

Section: Introductionmentioning

confidence: 95%

A quasi-isometric embedding theorem for groups

Olshanskii¹,

Osin²

2013

Duke Math. J.

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We show that every group H of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group G such that G is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever H is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups. arXiv:1202.6437v4 [math.GR] 30 Apr 2013In particular, our embedding allows us to carry over a wide range of properties from the group H to the group G. For instance, we have the following.Corollary 1.2. In the notation of Theorem 1.1, if H is solvable (respectively, satisfies a non-trivial identity, elementary amenable, amenable, has property A, has finite decomposition complexity, uniformly embeds in a Hilbert space, etc.), then so is G.Recall that property A was introduced by Yu [30] and groups of finite decomposition complexity were introduced by Guentner, Tessera, and Yu [16] with motivation coming from the Novikov conjecture and topological rigidity of manifolds, respectively. For definitions, properties, and applications we refer to [16,25,30] and references therein. Groups uniformly embeddable in Hilbert spaces are discussed below.The classes of elementary amenable groups, amenable groups, countable groups with property A, countable groups of finite decomposition complexity, and countable groups uniformly embeddable in a Hilbert space contain abelian groups and are closed with respect to

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Relative subgroup growth and subgroup distortion

Cited by 9 publications

References 2 publications

Computability of Følner sets

Computability of Følner sets

Some degree and distance-based invariants of wreath products of graphs

A quasi-isometric embedding theorem for groups

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