On the concept of “largeness” in group theory

Thomas, Simon

doi:10.1016/j.jalgebra.2009.08.019

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…Unlike some other natural preorders, such as "being a subgroup", "being a quotient group", or "being larger" in the sense of Pride (G Á P H if H 1 is a quotient of G 1 , for respective quotients G 1 , H 1 of finite-index subgroups of G, H by finite normal subgroups, see [53,58]), the preorder that we consider in this paper has infinitely many connected components. An easily described component is the connected component of Z: it contains all infinite abelian groups, and we describe the group of the order preserving bijections of this component in Proposition 3.7.…”

Section: The Structure Of Componentsmentioning

confidence: 99%

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Ordering the space of finitely generated groups

Bartholdi¹,

Erschler²

2015

Ann. inst. Fourier

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We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in (G,S_i) and in (H,T) coincide. Given a nilpotent group G, we characterize its connected component in this graph: if that connected component contains at least one torsion-free group, then it consists of those groups which generate the same variety of groups as G. The arrows in the graph define a preorder on the set of isomorphism classes of finitely generated groups. We show that a partial order can be imbedded in this preorder if and only if it is realizable by subsets of a countable set under inclusion. We show that every countable group imbeds in a group of non-uniform exponential growth. In particular, there exist groups of non-uniform exponential growth that are not residually of subexponential growth and do not admit a uniform imbedding into Hilbert space

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Section: The Structure Of Componentsmentioning

confidence: 99%

“…Thomas studies in [58] the complexity, with respect to the Borelian structure on G , of Pride's "largeness" preorder and of the "being a quotient" preorder. He shows that these preorders are high in the Borel hierarchy (they are what is called K σ -universal).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%