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Lattice subgroups of Kac-Moody groups
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Lattice subgroups of Kac-Moody groups
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ETD_1921
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000051797
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eng
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theses
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Mathematics
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Group theory
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Lattice theory
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(authority = ETD-LCSH)
Topic
Kac-Moody algebras
Abstract
We utilize graphs of groups and the corresponding covering theory to study lattices in type-infinity Kac-Moody groups over a finite field of size q, including results for both cocompact and nonuniform lattices.
For every prime power q we give a sufficient condition for the rank 2 Kac-Moody group G to contain a cocompact lattice with quotient a simplex, and we show that this condition is satisfied when q is a power of 2. Under further restrictions, we show that there is an infinite descending chain of cocompact lattices, and we demonstrate such a chain for q=2. Moreover we characterize the quotient graphs of groups for each lattice. Our method involves extending coverings of edge-indexed graphs to covering morphisms of graphs of groups. We also show how this gives rise to other infinite families of cocompact lattices in G.
When q=2 we are also able to embed the infinite descending chain in the rank 3 Kac-Moody group as a chain of lattices in the subgroup generated by all non-maximal standard parabolic subgroups. In addition we embed a non-discrete subgroup in the rank 3 Kac-Moody group whose quotient is a simplex.
We next give graphs of groups descriptions for known nonuniform lattices of Nagao-type. For the nonuniform lattices SL_2 and PGL_2 over polynomial rings with base field F_q we use the theory of ramified coverings to construct the graphs of groups for their congruence subgroups. We also examine the same construction employed by Morgenstern, identifying and repairing an error in his work. All graphs of groups for non-uniform lattices constructed here satisfy the structure theorem for graphs of groups.
For every prime power q we give a sufficient condition for the rank 2 Kac-Moody group G to contain a cocompact lattice with quotient a simplex, and we show that this condition is satisfied when q is a power of 2. Under further restrictions, we show that there is an infinite descending chain of cocompact lattices, and we demonstrate such a chain for q=2. Moreover we characterize the quotient graphs of groups for each lattice. Our method involves extending coverings of edge-indexed graphs to covering morphisms of graphs of groups. We also show how this gives rise to other infinite families of cocompact lattices in G.
When q=2 we are also able to embed the infinite descending chain in the rank 3 Kac-Moody group as a chain of lattices in the subgroup generated by all non-maximal standard parabolic subgroups. In addition we embed a non-discrete subgroup in the rank 3 Kac-Moody group whose quotient is a simplex.
We next give graphs of groups descriptions for known nonuniform lattices of Nagao-type. For the nonuniform lattices SL_2 and PGL_2 over polynomial rings with base field F_q we use the theory of ramified coverings to construct the graphs of groups for their congruence subgroups. We also examine the same construction employed by Morgenstern, identifying and repairing an error in his work. All graphs of groups for non-uniform lattices constructed here satisfy the structure theorem for graphs of groups.
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electronic resource
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x, 89 p. : ill.
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Ph.D.
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Includes bibliographical references (p. 86-88)
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by Ila Leigh Cobbs
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Cobbs
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Ila Leigh
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1980-
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Ila Leigh Cobbs
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Carbone
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Lisa
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chair
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Lisa Carbone
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Lepowsky
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James
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James Lepowsky
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Wilson
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Robert
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Robert Lee Wilson
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Kahrobaei
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Dalaram
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Dalaram Kahrobaei
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Rutgers University
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Graduate School - New Brunswick
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2009
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2009-10
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xx
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Rutgers University Electronic Theses and Dissertations
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ETD
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T3H70G0Z
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ETD doctoral
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The author owns the copyright to this work
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Open
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Cobbs
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Ila
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Ila Cobbs
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Rutgers University. Graduate School - New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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