Thesis (Ph.D.)--Tufts University, 2001. / Advisers: Mauricio Gutierrez; Zbigniew Nitecki. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 73-74). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph. D.)--University of Washington, 1992. / Vita. Includes bibliographical references (leaves [104]-105).

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2005. / Includes bibliographical references (p. 98-99).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (MSc)--University of Stellenbosch, 2005. / Bibliography. Also available via the Internet.

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Thesis (MSc (Mathematics))--University of Stellenboswch, 2010. / ENGLISH ABSTRACT: In the work of Green and Matignon it was shown that the Oort-Sekiguchi conjecture is equivalent to a local question of lifting automorphisms of power series rings. The Oort-Sekiguchi conjecture asks when an algebraic curve in characteristic p can be lifted to a relative curve in characteristic 0, while keeping the same automorphism group. The local formulation asks when an automorphism of a power series ring over a field k of characteristic p can be lifted to an automorphism of a power series ring over a discrete valuation ring with residue field k of the same order as the original automorphism. This thesis looks at the local formulation and surveys many of the results for this case. At the end it presents a new theorem giving a Hensel's Lemma type sufficient condition under which lifting is possible. / AFRIKAANSE OPSOMMING: Green en Matignon het bewys dat die Oort-Sekiguchi vermoede ekwivalent is aan `n lokale vraag oor of outomorfismes van magsreeksringe gelig kan word. Die Oort-Sekiguchi vermoede vra of `n algebra ese kromme in karakteristiek p gelig kan word na `n relatiewe kromme in karakteristiek 0, terwyl dit dieselfde outomorfisme groep behou. Die lokale vraag vra wanneer `n outomorfisme van `n magsreeksring oor `n liggaam k van karakteristiek p gelig kan word na `n outomorfisme van `n magsreeksring oor `n diskrete waarderingsring met residuliggaam k, terwyl dit dieselfde orde behou as die aanvanklike outomorfisme. Hierdie tesis fokus op die lokale vraag en bied `n opsomming van baie bekende resultate vir hierdie geval. Aan die einde word `n nuwe stelling aangebied wat voorwaardes stel waaronder hierdie vraag positief beantwoord kan word.

For various values of n and m we investigate homomorphisms from Out(F_n) to Out(F_m) and from Out(F_n) to GL_m(K), i.e. the free and linear representations of Out(F_n) respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(F_n) we prove that each homomorphism from Out(F_n) to GL_m(K) factors through the natural map p_n from Out(F_n) to GL(H_1(F_n,Z)) = GL_n(Z) whenever n=3, m < 7 and char(K) is not an element of {2,3}, and whenever n>5, m< n(n+1)/2 and char(K) is not an element of {2,3,...,n+1}. We also construct a new infinite family of linear representations of Out(F_n) (where n > 2), which do not factor through p_n. When n is odd these have the smallest dimension among all known representations of Out(F_n) with this property. Using the above results we establish that the image of every homomorphism from Out(F_n) to Out(F_m) is finite whenever n=3 and n < m < 6, and of cardinality at most 2 whenever n > 5 and n < m < n(n-1)/2. We further show that the image is finite when n(n-1)/2 -1 < m < n(n+1)/2. We also consider the structure of normal finite index subgroups of Out(F_n). If N is such then we prove that if the derived subgroup of the intersection of N with the Torelli subgroup T_n < Out(F_n) contains some term of the lower central series of T_n then the abelianisation of N is finite.

How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.

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