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Negligibity of elliptic elements in ascending HNN-extensions of

Negligibity of elliptic elements in ascending HNN-extensions of
Negligibity of elliptic elements in ascending HNN-extensions of

We study ascending HNN-extensions G of finitely generated free abelian groups: examples of such G include soluble Baumslag–Solitar groups and fundamental groups of orientable prime 3-manifolds modeled on Sol geometry. In particular, we study the elliptic subgroup A ≤ G consisting of all elements that stabilize a point in the Bass–Serre tree of G. We consider the density of A with respect to ball counting measures corresponding to finite generating sets of G, and we show that A is exponentially negligible in G with respect to such sequences of measures. As a consequence, we show that the set of tuples (x0, ...,xr) εGr+1 such that the (r+1) -fold simple commutator [x0, ...,xr] vanishes, is exponentially negligible in Gr+1 with respect to sequences of ball counting measures.

Ascending HNN-extensions, Baumslag–Solitar groups, degree of nilpotence, elliptic elements, Sol-manifolds
0092-7872
1-20
Valiunas, Motiejus
23b32cdf-14c1-409c-9fc7-d53b54623233
Valiunas, Motiejus
23b32cdf-14c1-409c-9fc7-d53b54623233

Valiunas, Motiejus (2019) Negligibity of elliptic elements in ascending HNN-extensions of. Communications in Algebra, 1-20. (doi:10.1080/00927872.2019.1612417).

Record type: Article

Abstract

We study ascending HNN-extensions G of finitely generated free abelian groups: examples of such G include soluble Baumslag–Solitar groups and fundamental groups of orientable prime 3-manifolds modeled on Sol geometry. In particular, we study the elliptic subgroup A ≤ G consisting of all elements that stabilize a point in the Bass–Serre tree of G. We consider the density of A with respect to ball counting measures corresponding to finite generating sets of G, and we show that A is exponentially negligible in G with respect to such sequences of measures. As a consequence, we show that the set of tuples (x0, ...,xr) εGr+1 such that the (r+1) -fold simple commutator [x0, ...,xr] vanishes, is exponentially negligible in Gr+1 with respect to sequences of ball counting measures.

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More information

Accepted/In Press date: 14 April 2019
e-pub ahead of print date: 18 May 2019
Keywords: Ascending HNN-extensions, Baumslag–Solitar groups, degree of nilpotence, elliptic elements, Sol-manifolds

Identifiers

Local EPrints ID: 432908
URI: http://eprints.soton.ac.uk/id/eprint/432908
ISSN: 0092-7872
PURE UUID: 97cbb498-39c7-4174-89bb-268a176cb1f4
ORCID for Motiejus Valiunas: ORCID iD orcid.org/0000-0003-1519-6643

Catalogue record

Date deposited: 31 Jul 2019 16:30
Last modified: 01 Aug 2019 00:39

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