Random subgroups of Thompson's group F

Cleary, S; Elder, M; Rechnitzer, A; Taback, J

Random subgroups of Thompson's group F

Cleary, S Elder, M

Rechnitzer, A Taback, J

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Publication Type:: Journal Article
Citation:: Groups, Geometry, and Dynamics, 2010, 4 (1), pp. 91 - 126
Issue Date:: 2010-01-13

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Field	Value	Language
dc.contributor.author	Cleary, S	en_US
dc.contributor.author	Elder, M https://orcid.org/0000-0002-2438-3945	en_US
dc.contributor.author	Rechnitzer, A	en_US
dc.contributor.author	Taback, J	en_US
dc.date.issued	2010-01-13	en_US
dc.identifier.citation	Groups, Geometry, and Dynamics, 2010, 4 (1), pp. 91 - 126	en_US
dc.identifier.issn	1661-7207	en_US
dc.identifier.uri	http://hdl.handle.net/10453/98849
dc.description.abstract	We consider random subgroups of Thompson's group F with respect to two natural stratifications of the set of all k-generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of persistent subgroups, whose isomorphism classes occur with positive density within the set of k-generator subgroups, for all sufficiently large k. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of F are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite (short for differentiably finite) and not algebraic. We then use the asymptotic growth to prove our density results. © European Mathematical Society.	en_US
dc.relation.ispartof	Groups, Geometry, and Dynamics	en_US
dc.relation.isbasedon	10.4172/GGD/76	en_US
dc.title	Random subgroups of Thompson's group F	en_US
dc.type	Journal Article
utslib.citation.volume	1	en_US
utslib.citation.volume	4	en_US
utslib.for	0101 Pure Mathematics	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Science
pubs.organisational-group	/University of Technology Sydney/Faculty of Science/School of Mathematical and Physical Sciences
utslib.copyright.status	open_access
pubs.issue	1	en_US
pubs.publication-status	Published	en_US
pubs.volume	4	en_US

Abstract:

We consider random subgroups of Thompson's group F with respect to two natural stratifications of the set of all k-generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of persistent subgroups, whose isomorphism classes occur with positive density within the set of k-generator subgroups, for all sufficiently large k. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of F are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite (short for differentiably finite) and not algebraic. We then use the asymptotic growth to prove our density results. © European Mathematical Society.

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http://hdl.handle.net/10453/98849