Thesis (Ph. D.)--University of Wisconsin--Madison, 1963. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 90-91).

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Let G be a finite group. Given a contravariant, product preserving functor F:G-sets → AB, we construct a Green-functor A(F):G-sets → CRNG which specializes to the Burnside ring functor when F is trivial. A(F) permits a natural addition and multiplication between elements in the various groups F(S), S ∈ G-sets. If G is the Galois group of a field extension L/K, and SEP denotes the category of K-algebras which are isomorphic with a finite product of subfields of L, then any covariant, product preserving functor ρ:SEP → AB induces a functor Fᵨ:G → AB, and thus the Green-functor Aᵨ may be obtained. We use this observation for the case ρ = Br, the Brauer group functor, and show that Aᵦᵣ(G/G) is free on K-algebra isomorphism classes of division algebras with center in SEP. We then interpret the induction theory of Mackey-functors in this context. For a certain class of functors F, the structure of A(F) is especially tractable; for these functors we deduce that (DIAGRAM OMITTED), where the product is over isomorphism class representatives of transitive G-sets. This allows for the computation of the prime ideals of A(F)(G/G), and for an explicit structure theorem for Aᵦᵣ, when G is the Galois group of a p-adic field. We finish by considering the case when G = Gal(L/Q), for an arbitrary number field L.

This thesis -- originally entitled “Reckoning the Unnamed Fabric”, both a cultural study of the poetics of ecological consciousness and the ecology of poetic consciousness -- investigates the post-Romantic legacy informing John Burnside’s (b. 1955) poetry from The hoop (1988) to The Light Trap (2002) as a case study. The thesis argues that a developing aesthetic form and movement in subject derive from Burnside’s increasing involvement with ecological thought and practice. This move to the poetry of the oikos begins with an investigation of the self through the reconciliation of subject with object (or human with nature), and latterly has moved into a sustained reflection upon the idea of dwelling. This thesis relates the chronological development across Burnside’s nature poetry to an aesthetic infused with religious iconography and language, which via an evolving motif-poem of ‘world-soul’ or ‘communal fabric’ increases in its secular and empirical inflection. I read Burnside’s elevation of historical materialism s a progression in Wordsworthian craft and as a result of the poet’s pragmatic reflection on dwelling; I argue that the poetic consolidation of the intrinsic value of nature as an active and guiding spirit promotes nature less as a place for inhabitants than as the site and point of relation. The argument responds to Burnside’s transatlantic perspective from which he questions what it means to live as a spirit, and what a poetics of ecology can achieve in respect to the human subjective lyric and the need to transcend the human into the collective. To address these questions, which are implicit in Burnside's oeuvre, I draw upon Heideggerian poetics and American post-Transcendentalist Romanticism. I locate Burnside’s poetics within philosophical, aesthetic, and ecological frameworks. First, Burnside’s poetry is primarily a poetics of ontology that understands the ‘I’ within the midst of things yet underpinned by epistemology/hermeneutics; second, Burnside exhibits neo-Romantic poetry that has engaged with Modern American poetry -- it is this fusion that I call post-Romantic; third, the ecological constitutes both Burnside’s political stance and his aesthetic-poetic stance. I read the latter as a reflection of Jonathan Bate’s notion of the ecopoem as the “post-phenomenological inflection of high Romantic poetics”, an idea which is most apposite when read in relationship with Burnside’s path towards the metaphysical inscribed in the historical.

Stuttgart, Univ., Diss., 2008.

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Making use of linear and homological algebra techniques we study the linearization map between the generalized Burnside and rational representation rings of a group G. For groups G and H, the generalized Burnside ring is the Grothendieck construction of the semiring of G × H-sets with a free H-action. The generalized representation ring is the Grothendieck construction of the semiring of rational G×H-modules that are free as rational H-modules. The canonical map between these two rings mapping the isomorphism class of a G-set X to the class of its permutation module is known as the linearization map. For p a prime number and H the unique group of order p, we describe the generators of the kernel of this map in the cases where G is an elementary abelian p-group or a cyclic p-group. In addition we introduce the methods needed to study the Bredon homology theory of a G-CW-complex with coefficients coming from the classical Burnside ring.

Le groupe de Burnside libre d'exposant n, B(r,n), est le quotient du groupe libre de rang r par le sous-groupe engendré par les puissance n-ièmes de tous ses éléments. Ce groupe fut introduit en 1902 par W. Burnside qui demandait si un tel objet était nécessairement fini. Depuis les travaux de P.S. Novikov et S.I. Adian à la fin des années soixante, on sait que, pour des exposants suffisamment grands, la réponse est négative. Dans cette thèse on s'intéresse aux automorphismes extérieurs de B(r,n). En adaptant l'approche géométrique de la théorie de la petite simplification développée par T. Delzant et M. Gromov, on exhibe une large classe d'automorphismes du groupe libre qui induisent des éléments d'ordre infini de Out(B(r,n)). On montre aussi que Out(B(r,n)) contient des sous-groupes libres et abéliens libres.

This dissertation is devoted to various aspects of groups generated by automata. We study particular classes and examples of such groups from different points of view. It consists of four main parts. In the first part we study Sushchansky p-groups introduced in 1979 by Sushchansky in "Periodic permutation p-groups and the unrestricted Burnside problem". These groups represent one of the earliest examples of Burnside groups and, at the same time, show the potential of the class of groups generated by automata to contain groups with extraordinary properties. The original definition is translated into the language of automata. The original actions of Sushchansky groups on p- ary tree are not level-transitive and we describe their orbit trees. This allows us to simplify the definition and prove that these groups admit faithful level-transitive actions on the same tree. Certain branch structures in their self-similar closures are established. We provide the connection with so-called G groups introduced by Bartholdi, Grigorchuk and Suninc in "Branch groups" that shows that all Sushchansky groups have intermediate growth and allows us to obtain an upper bound on their period growth functions. The second part is devoted to the opposite question of realization of known groups as groups generated by automata. We construct a family of automata with n states, n greater than or equal to 4, acting on a rooted binary tree and generating the free products of cyclic groups of order 2. The iterated monodromy group IMG(z2+i) of the self-map of the complex plain z -> z2 + i is the central object of the third part of dissertation. This group acts faithfully on the binary rooted tree and is generated by 4-state automaton. We provide a self-similar measure for this group giving alternative proof of its amenability. We also compute an L-presentation for IMG(z2+i) and provide calculations related to the spectrum of the Markov operator on the Schreier graph of the action of IMG(z2 + i) on the orbit of a point on the boundary of the binary rooted tree. Finally, the last part is discussing the package AutomGrp for GAP system developed jointly by the author and Yevgen Muntyan. This is a very useful tool for studying the groups generated by automata from the computational point of view. Main functionality and applications are provided.

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