One-relator groups with torsion, virtually free-by-cyclic groups, and free-by-free groups.
Item
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Title
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One-relator groups with torsion, virtually free-by-cyclic groups, and free-by-free groups.
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Identifier
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AAI9130360
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identifier
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9130360
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Creator
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Persinger, Sharon E.
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Contributor
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Adviser: Gilbert Baumslag
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Date
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1991
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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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Gilbert Baumslag has conjectured that any one-relator group with torsion is virtually free-by-cyclic. This work has its origin in an attempt to prove that conjecture, and so show that any one-relator group with torsion is necessarily Hopfian.;We consider some generalizations of the class of free groups which arise in the analysis of one-relator groups, and examine some properties these groups share with free groups. The properties we will examine include residual finiteness and Hopficity.;By explicit computations we show that certain one relator groups with torsion are virtually free-by-cyclic. Specifically, we prove the theorems:;Theorem. The groups {dollar}G\sb{lcub}p,q,k{rcub} = \langle x,y; (x\sp{lcub}-1{rcub}y\sp{lcub}p{rcub}xy\sp{lcub}q{rcub})\sp{lcub}k{rcub}\rangle,{dollar} where {dollar}0 < p,q < k,{dollar} and {dollar}(p + q,k) = 1,{dollar} are virtually free-by-cyclic.;Theorem. The groups {dollar}H\sb{lcub}m{rcub} = \langle a,b; \lbrack a, b\rbrack\sp{lcub}m{rcub}\rangle{dollar} are virtually free-by-cyclic.;Theorem. The groups {dollar}S\sb{lcub}k{rcub} = \langle a,b,g\sb{lcub}i{rcub}, i \in I; (\lbrack a,b\rbrack w)\sp{lcub}k{rcub}{dollar}, w a word in {dollar}g\sb{lcub}i{rcub}\rangle{dollar} are virtually free-by-cyclic.;In addition we show:;Theorem. The groups {dollar}G\sb{lcub}p,q,k{rcub}{dollar} and {dollar}H\sb{lcub}m{rcub}{dollar} are not virtually free.;We also examine closure properties in the class of virtually free-by-cyclic groups, leading up to two results of this type:;Theorem. Suppose A and B are virtually free-by-cyclic groups: A is a finite extension of {dollar}M =\langle x,F\sb1\rangle{dollar} and B is a finite extension of {dollar}N = \langle y,F\sb2\rangle.{dollar} Let {dollar}G = A{lcub}*\atop x=y{rcub}B{dollar} be their free product amalgamating {dollar}\langle x\rangle{dollar} and {dollar}\langle y\rangle{dollar}. Then, under some extra conditions on the conjugates of x in A and the conjugates of y in B, G is virtually free-by-cyclic.;The more general class of free-by-free groups is considered next. We look at the kernel of a homomorphism from a one-relator group onto a free group, and prove several theorems which describe conditions on the relator under which this kernel will be free.;In the class of virtually residually free groups, we first show some basic results involving closure properties. The main result is:;Theorem. Let A and B be finitely generated abelian groups and {dollar}G = A{lcub}*\atop C{rcub}B.{dollar} Then G is virtually residually free.
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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.
