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University of Oklahoma

1. Wright, Rachel. Totally Reflected Groups.

Degree: PhD, 2016, University of Oklahoma

URL: http://hdl.handle.net/11244/34633 

A group G is totally reflected if it has a generating set S such that each edge in the Cayley graph Gamma(G,S) is inverted by some color-preserving reflection on the graph. For example, we will show that Coxeter groups and right-angled Artin groups are totally reflected and that a finitely generated abelian group is totally reflected if and only if its first invariant factor is even. We show that direct and free products of totally reflected groups are totally reflected. More generally, we develop a group construction called a right-angled product which generalizes free and direct products, and we show that a right-angled product of totally reflected groups is itself totally reflected. A group G is strongly totally reflected if there exists a color-preserving reflection group G_R acting on Gamma(G,S) such that each edge in the graph is inverted by some reflection in G_R. We state and prove sufficient conditions for a totally reflected group to be strongly totally reflected and use these results to prove from a graphical perspective that any right-angled Artin group is commensurable with a right-angled Coxeter group. In particular, we show that both the right-angled Artin group A(Delta)=<S> and its associated right-angled Coxeter group A_r are finite-index subgroups of the group of color-preserving graph automorphisms of Gamma(A(Delta),S). Advisors/Committee Members: Miller, Andrew (advisor), Rubin, Leonard (committee member), Reeder, Stacy (committee member), Stewart, Sepideh (committee member), Ozaydin, Murad (committee member).

Subjects/Keywords: Mathematics.; graph reflections; right-angled product; Cayley graph; geometric group theory

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

APA (6th Edition):

Wright, R. (2016). Totally Reflected Groups. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/34633

Chicago Manual of Style (16th Edition):

Wright, Rachel. “Totally Reflected Groups.” 2016. Doctoral Dissertation, University of Oklahoma. Accessed March 09, 2021. http://hdl.handle.net/11244/34633.

MLA Handbook (7th Edition):

Wright, Rachel. “Totally Reflected Groups.” 2016. Web. 09 Mar 2021.

Vancouver:

Wright R. Totally Reflected Groups. [Internet] [Doctoral dissertation]. University of Oklahoma; 2016. [cited 2021 Mar 09]. Available from: http://hdl.handle.net/11244/34633.

Council of Science Editors:

Wright R. Totally Reflected Groups. [Doctoral Dissertation]. University of Oklahoma; 2016. Available from: http://hdl.handle.net/11244/34633


Universidade Estadual de Campinas

2. Talpo, Humberto Luiz. Reflexões e numero de cobertura de arvores homogeneas e grupos de automorfismos de arvores semi-homogeneas.

Degree: 2006, Universidade Estadual de Campinas

URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/305924 

Abstract: Let G be a homogeneous tree and Aut(G) its group of automorphism. An automorphism Î Aut(G) is said to be even if d(f(x),x) º0 mod 2 for every vertex x Î G of , where d(.,.) is the canonical distance function defined by the minimum length of paths connecting the vertices. The set Aut+(G) of all even automorphism is a subgroup of index 2 in Aut(G). We define a geodesic g Ì G as a subtree isomorphic to the standard tree of the integers Z, that is, a homogeneous subtree of degree 2. A reflection in a geodesic g is an involutive automorphism f (f² =1) such that f(x) = x if x Î G. We denote by R the set of all reflections in geodesics. In this work (Chapter 2) we prove that, for every even degree tree G, the covering number of Aut+(G) by reflections in geodesics is 11, in the sense that give f Î Aut+(G) there are f1, f2,... fk with k £ 11, such that f(x) = fk °fk-1°...°f1(x) for every vertex x in G.Moreover, if we consider homogeneous trees we know that automorphisms group is complete and the even automorphisms subgroup is simple. We vary the homogeneous condition and we prove that (Chapter 3) for the semi-homogeneous trees, the automorphisms group is simple and complete Advisors/Committee Members: UNIVERSIDADE ESTADUAL DE CAMPINAS (CRUESP), Firer, Marcelo, 1961- (advisor), San Martin, Luiz Antonio Barrera, 1955- (coadvisor), Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica (institution), Programa de Pós-Graduação em Matemática (nameofprogram), Vieira, Ana Cristina (committee member), Souza, Simone Dantas de (committee member), Carmelo, Emerson Luiz do Monte (committee member), Santos, José Plínio de Oliveira (committee member).

Subjects/Keywords: Reflexões; Árvores (Teoria dos grafos); Automorfismo; Isometria (Matemática); Reflections; Trees (Graph theory); Automorphism; Isometry (Mathematics)

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

APA (6th Edition):

Talpo, H. L. (2006). Reflexões e numero de cobertura de arvores homogeneas e grupos de automorfismos de arvores semi-homogeneas. (Thesis). Universidade Estadual de Campinas. Retrieved from http://repositorio.unicamp.br/jspui/handle/REPOSIP/305924

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Talpo, Humberto Luiz. “Reflexões e numero de cobertura de arvores homogeneas e grupos de automorfismos de arvores semi-homogeneas.” 2006. Thesis, Universidade Estadual de Campinas. Accessed March 09, 2021. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305924.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Talpo, Humberto Luiz. “Reflexões e numero de cobertura de arvores homogeneas e grupos de automorfismos de arvores semi-homogeneas.” 2006. Web. 09 Mar 2021.

Vancouver:

Talpo HL. Reflexões e numero de cobertura de arvores homogeneas e grupos de automorfismos de arvores semi-homogeneas. [Internet] [Thesis]. Universidade Estadual de Campinas; 2006. [cited 2021 Mar 09]. Available from: http://repositorio.unicamp.br/jspui/handle/REPOSIP/305924.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Talpo HL. Reflexões e numero de cobertura de arvores homogeneas e grupos de automorfismos de arvores semi-homogeneas. [Thesis]. Universidade Estadual de Campinas; 2006. Available from: http://repositorio.unicamp.br/jspui/handle/REPOSIP/305924

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.