On the structure of the space of lattices in a class of simply connected, 2-step solvable real Lie groups and genus sets of certain spaces.
Item
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Title
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On the structure of the space of lattices in a class of simply connected, 2-step solvable real Lie groups and genus sets of certain spaces.
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Identifier
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AAI9807943
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identifier
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9807943
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Creator
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Huang, Huale.
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Contributor
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Advisers: Martin Moskowitz | Joseph Roitberg
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Date
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1997
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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 classify all the lattices for a class of 2-step solvable simply connected Lie groups G. These are semi-direct products of {dollar}\IR{dollar} acting on {dollar}\IR\sp{lcub}n{rcub}{dollar} via a 1-parameter subgroup whose infinitesimal generator, {dollar}\Delta,{dollar} is real upper triangular and of trace zero. We show that the lattices of G are all of the forms {dollar}{lcub}\cal L{rcub}(A,\sigma),{dollar} where {dollar}A\in {lcub}\rm SL{rcub}(n,\doubz),\sigma\in {lcub}\rm GL{rcub}(n,\IR){dollar} and {dollar}\sigma\sp{lcub}-1{rcub}A\sigma{dollar} equals exp {dollar}t\Delta{dollar} for some real {dollar}t\not=0.{dollar} The lattice {dollar}{lcub}\cal L{rcub}(A,\sigma){dollar} is then given by {dollar}\sigma\sp{lcub}-1{rcub}\doubz\sp{lcub}n{rcub}\times t\doubz.{dollar} Furthermore, two such lattices {dollar}{lcub}\cal L{rcub}(A,\sigma){dollar} and {dollar}{lcub}\cal L{rcub}(B,\tau){dollar} differ by a smooth automorphism of G if and only if A and B are extendedly conjugate. We then turn to some questions concerning the decomposition of the quasi-regular representation for such groups. We show that when {dollar}n=2{dollar} the representation decomposes into a direct sum of indecomposable subrepresentations in such a way that although each of these subrepresentation occurs with finite times, the multiplicity function itself is always unbounded. Then we turn to compute genus sets for quaternion projective spaces and wedges of finitely many spheres of the same dimension. In fact, we prove that both the Mislin genus set {dollar}{lcub}\cal G{rcub}(HP\sp{lcub}n{rcub}){dollar} and the genus set {dollar}\{lcub}\cal G{rcub}\sb0(S\sp{lcub}n{rcub}\vee S\sp{lcub}n{rcub}){dollar} (where {dollar}n>2){dollar} are uncountably large.
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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.
