Random subgroups of Thompson’s group $F$
Copyright policy )Random subgroups of Thompson’s group $F$
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.
- University of Queensland Australia
- King’s University United States
- University of Queensland Australia
- University of Newcastle Australia Australia
- Department of Mathematics University of British Columbia Canada
non-algebraic generating function, D-finite generating functions, 05A05, 20F65, asymptotic densities, statistical group theory, visible subgroup, Exact enumeration problems, generating functions, Subgroup theorems; subgroup growth, subgroup spectra, Thompson's group F, Group Theory (math.GR), 510, Trees, Probabilistic methods in group theory, FOS: Mathematics, Non-algebraic generating function, Mathematics - Combinatorics, Persistent subgroup, D-finite generating function, asymptotic density, visible subgroups, 0105 Mathematical Physics, Visible subgroup, asymptotic group theory, Subgroup spectrum, Statistical group theory, Graphs and abstract algebra (groups, rings, fields, etc.), persistent subgroup, Asymptotic group theory, subgroup spectrum, Asymptotic density, Asymptotic properties of groups, Thompson group \(F\), non-algebraic generating functions, Combinatorics (math.CO), Geometric group theory, persistent subgroups, Mathematics - Group Theory
non-algebraic generating function, D-finite generating functions, 05A05, 20F65, asymptotic densities, statistical group theory, visible subgroup, Exact enumeration problems, generating functions, Subgroup theorems; subgroup growth, subgroup spectra, Thompson's group F, Group Theory (math.GR), 510, Trees, Probabilistic methods in group theory, FOS: Mathematics, Non-algebraic generating function, Mathematics - Combinatorics, Persistent subgroup, D-finite generating function, asymptotic density, visible subgroups, 0105 Mathematical Physics, Visible subgroup, asymptotic group theory, Subgroup spectrum, Statistical group theory, Graphs and abstract algebra (groups, rings, fields, etc.), persistent subgroup, Asymptotic group theory, subgroup spectrum, Asymptotic density, Asymptotic properties of groups, Thompson group \(F\), non-algebraic generating functions, Combinatorics (math.CO), Geometric group theory, persistent subgroups, Mathematics - Group Theory
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