Dissertation/Thesis Abstract

Asymptotic Geometry, Bounded Generation and Subgroups of Mapping Class Groups
by Sun, Chun-yi, Ph.D., Yale University, 2012, 64; 3525282
Abstract (Summary)

A group G is boundedly generated (by cyclic groups) if there is a finite ordered set (gi[special characters omitted], such that G = {[special characters omitted]}. Let (gi[special characters omitted] be a finite sequence of elements in [special characters omitted](S). For any word [special characters omitted] without obvious cancellation, we can estimate its stable length with [special characters omitted]|ni|. We demonstrate that if a subgroup of [special characters omitted](S) has exponential growth, it cannot be boundedly generated. Combining this with the Tits alternative for mapping class groups, we derive that a subgroup of [special characters omitted](S) is boundedly generated if and only if it is virtually abelian. Suppose further that each gi is non-elliptic, and that any gi and gj are not commensurable up to conjugacy if i j. We apply the same estimate to show that there exists m > 0 such that normal subgroup ⟨⟨[special characters omitted]|1 ≤ ik⟩⟩ is an infinitely generated right-angled Artin group provided n is sufficiently large.

We use a different method to show that if G < [special characters omitted](S) has exponential growth, G cannot be boundedly generated by cyclic subgroups and/or curve stabilizers. That is G ≠ Γ1···Γ k if each Γi is either cyclic or a curve stabilizer.

Indexing (document details)
Advisor: Minsky, Yair
School: Yale University
School Location: United States -- Connecticut
Source: DAI-B 73/12(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Asymptotic geometries, Bounded generation, Mapping class groups
Publication Number: 3525282
ISBN: 978-1-267-57399-5
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