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University of Oklahoma

1.
Sulman, Robert M.
*Affine**group* actions on Euclidean space.

Degree: PhD, Department of Mathematics, 2006, University of Oklahoma

URL: http://hdl.handle.net/11244/1016

Upto affine conjugacy, we describe properly discontinuous rank two affine groups of Euclidean space (primarily dimensions two and three) in terms of "coordinates" of generators. In dimension two, a chosen generator is put into a normal form. The commuting condition simplifies a second generator, which can be identified with a point of the plane. Thus, R2 can be viewed as a parameter space of groups (since the first normalized generator is common to each group). A homomorphism Res:R2->R (the residue) singles out properly discontinuous groups G isomorphic to Z+Z. Affine conjugacy of two groups is characterized by their residues and an element of GL (2, Z). As a consequence, we show (i) There are uncountably many conjugacy classes of properly discontinuous rank two groups, and (ii) Each point of Ker(Res) is the limit point of every conjugacy class. An analog of the residue is used to determine three dimensional properly discontinuous rank two groups, although this description does not cover all normalized forms.
*Advisors/Committee Members: Basmajian, Ara, (advisor).*

Subjects/Keywords: Group schemes (Mathematics); Mathematics.; Affine algebraic groups.

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Sulman, R. M. (2006). Affine group actions on Euclidean space. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/1016

Chicago Manual of Style (16^{th} Edition):

Sulman, Robert M. “Affine group actions on Euclidean space.” 2006. Doctoral Dissertation, University of Oklahoma. Accessed September 21, 2020. http://hdl.handle.net/11244/1016.

MLA Handbook (7^{th} Edition):

Sulman, Robert M. “Affine group actions on Euclidean space.” 2006. Web. 21 Sep 2020.

Vancouver:

Sulman RM. Affine group actions on Euclidean space. [Internet] [Doctoral dissertation]. University of Oklahoma; 2006. [cited 2020 Sep 21]. Available from: http://hdl.handle.net/11244/1016.

Council of Science Editors:

Sulman RM. Affine group actions on Euclidean space. [Doctoral Dissertation]. University of Oklahoma; 2006. Available from: http://hdl.handle.net/11244/1016

University of Southern California

2. Warner, Harry Jared, IV. Springer isomorphisms and the variety of elementary subalgebras.

Degree: PhD, Mathematics, 2015, University of Southern California

URL: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/541250/rec/6023

Over a field of large enough characteristic, we use
the canonical Springer isomorphism between the unipotent variety of
a connected, reductive group and the nilpotent variety of the
group's Lie algebra to study the projective variety of elementary
subalgebras, as defined by Jon Carlson, Eric Friedlander, and Julia
Pevtsova. When our structures are defined over finite fields, we
relate certain computable information about elementary abelian
subgroups of Chevalley groups with rational points and orbits of
the variety of elementary subalgebras. We also construct the
commuting variety of one-parameter subgroups which simultaneously
generalizes the variety of elementary subalgebras and the variety
of one-parameter subgroups, as defined by Andrei Suslin, Eric
Friedlander, and Christopher Bendel. We employ Magma to make many
computations and draw certain visualizations of the
theory.
*Advisors/Committee Members: Friedlander, Eric M. (Committee Chair), Guralnick, Robert M. (Committee Member), Montgomery, M. Susan (Committee Member), Jonckheere, Edmond A. (Committee Member).*

Subjects/Keywords: affine group schemes; representation theory; support varieties; Springer isomorphism; algebraic groups; elementary subalgebras; restricted Lie algebras; elementary Abelian subgroups

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Warner, Harry Jared, I. (2015). Springer isomorphisms and the variety of elementary subalgebras. (Doctoral Dissertation). University of Southern California. Retrieved from http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/541250/rec/6023

Chicago Manual of Style (16^{th} Edition):

Warner, Harry Jared, IV. “Springer isomorphisms and the variety of elementary subalgebras.” 2015. Doctoral Dissertation, University of Southern California. Accessed September 21, 2020. http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/541250/rec/6023.

MLA Handbook (7^{th} Edition):

Warner, Harry Jared, IV. “Springer isomorphisms and the variety of elementary subalgebras.” 2015. Web. 21 Sep 2020.

Vancouver:

Warner, Harry Jared I. Springer isomorphisms and the variety of elementary subalgebras. [Internet] [Doctoral dissertation]. University of Southern California; 2015. [cited 2020 Sep 21]. Available from: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/541250/rec/6023.

Council of Science Editors:

Warner, Harry Jared I. Springer isomorphisms and the variety of elementary subalgebras. [Doctoral Dissertation]. University of Southern California; 2015. Available from: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/541250/rec/6023