Conjugately dense subgroups of free products of groups with amalgamation

Abstract

A subgroup having non-empty intersection with each class of conjugate elements of the group is said to be conjugately dense. It is shown that, under certain conditions, the number of conjugately dense subgroups in a free product with amalgamation is not less than some cardinal. As a consequence, P. Neumann's conjecture in the Kourovka notebook (Question 6.38) is refuted. It is also stated that a modular group and a non-Abelian group of countable or finite rank possess continuum many pairwise non-conjugate conjugately dense subgroups.

Original language	English
Pages (from-to)	296-305
Number of pages	10
Journal	Algebra and Logic
Volume	45
Issue number	5
DOIs	https://doi.org/10.1007/s10469-006-0028-1
Publication status	Published - Sep 2006

Keywords

Conjugately dense subgroup
Field with discrete valuation
Free product with amalgamation
Linear group

ASJC Scopus subject areas

Algebra and Number Theory

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10.1007/s10469-006-0028-1

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abstract = "A subgroup having non-empty intersection with each class of conjugate elements of the group is said to be conjugately dense. It is shown that, under certain conditions, the number of conjugately dense subgroups in a free product with amalgamation is not less than some cardinal. As a consequence, P. Neumann's conjecture in the Kourovka notebook (Question 6.38) is refuted. It is also stated that a modular group and a non-Abelian group of countable or finite rank possess continuum many pairwise non-conjugate conjugately dense subgroups.",

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AB - A subgroup having non-empty intersection with each class of conjugate elements of the group is said to be conjugately dense. It is shown that, under certain conditions, the number of conjugately dense subgroups in a free product with amalgamation is not less than some cardinal. As a consequence, P. Neumann's conjecture in the Kourovka notebook (Question 6.38) is refuted. It is also stated that a modular group and a non-Abelian group of countable or finite rank possess continuum many pairwise non-conjugate conjugately dense subgroups.

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KW - Linear group

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