Metric properties of Thompson's groups F(n) and F(n,m).
Item
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Title
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Metric properties of Thompson's groups F(n) and F(n,m).
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Identifier
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AAI3283170
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identifier
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3283170
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Creator
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Wladis, Claire W.
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Contributor
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Adviser: Sean Cleary
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Date
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2007
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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We prove several metric properties of Thompson's group F( n) with respect to the standard finite generating set {lcub}x 0, x1,..., x p--1, xp{rcub}. We prove that seesaw words exist in F(n) with respect to {lcub}x0, x1,..., xp--1, xp{rcub} and therefore that F(n) is not combable by geodesics and that there does not exist a regular language of geodesics for F(n). We also prove that dead ends exist in F(n) with respect to {lcub}x 0, x1,..., xp --1, xp{rcub} and that these dead ends all have depth 2. Both these results were proven for the case n = 2 (i.e. when F(n) = F) by Cleary and Taback; our paper extends these results to F( n) for n = 2, 3, 4,.... We then prove that F(n) is not minimally almost convex with respect to {lcub}x0, x1,..., xp--1, xp{rcub}. Belk and Bux proved this for n = 1. We follow the outline of their proof, generalizing it to F(n) for all n ∈ {lcub}2, 3, 4,...{rcub} by representing elements of F(n) as tree-pair diagrams rather than forest diagrams in order to use Fordham's metric on F(n).;We also prove several metric properties of Thompson's group F(n,m) where n,m ∈ {lcub}2, 3, 4,...{rcub}. We highlight several differences between F( n) and F(n,m), including the fact that minimal tree-pair diagram representatives of elements of F( n,m) may not be unique. We establish how to find minimal tree-pair diagram representatives of elements of F(n,m) and give two different ways to find unique minimal representatives for elements of the group. We establish how tree-pair diagrams can be composed in order to perform group multiplication, and we prove several theorems describing the equivalence of trees and tree-pair diagrams. We introduce a normal form for elements of F(2, 3) with respect to the standard infinite generating set. We also give bounds for the metric of F(2, 3) with respect to the standard finite generating set and show that the order of these bounds is sharp, showing that the metric on F(2, 3) is not quasi-isometric to the number of leaves or carets in the minimal tree-pair diagram representative, as is the case in F(2).;Index words. Thompson's group, Higman group, Stein group.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
