Topological and Dynamical Properties of Cyclically Presented Groups Public Deposited

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  • This dissertation investigates the structure and topological properties of cyclicallypresented groups. First, a family of groups called groups of type Z is considered. Withfew exceptions, the finiteness, asphericity, fixed point, and 3-manifold spine problemsare solved. Most groups of type Z have a central element of infinite order fixed by theshift. Moreover, the shift extension decomposes as an amalgamated product, with factor acentrally extended triangle group. Consequently, most groups of type Z are SQ-universal.Heegaard diagrams are constructed to produce a class of Seifert fibered 3-manifolds; eachis a cyclic branched cover of a lens space. Generalizing those in [16], this class includesthe Brieskorn [50] and Sieradski manifolds [64].The shift dynamics of the groups Gn(k,m) = Gn(x0xmxk^-1) of Fibonacci type [40]are investigated. These generalize the Fibonacci groups F(2, n) = Gn(1, 2) [20], Sieradskigroups S(2, n) = Gn(2, 1) [64], and Gilbert- Howie groups H(n,m) = Gn(m; 1) [28].Modulo two unresolved cases, the asphericity problem for groups of Fibonacci type wassolved in [71]; finiteness [72]; 3-manifold group [37]. Here the main finding is the shiftaction is faithful for all n; m; k such that Gn(m; k) is nontrivial. The fixed point andfreeness problems are also solved, except for a handful of groups including H(9; 4) andH(9; 7). The first known examples of infinite, non-aspherical groups with free shift actionare discovered.Finally, the situation is considered when a word w 2 F(x0,...,xn-1) in a cyclicpresentation Pn(w) is composite, i.e. w = v u. We find the structure of Gn(v o u)depends on Gn(u) and Gn(v), with respect to the finiteness, asphericity, shift dynamics,and 3-manifold spine problems. The idea of studying Gn(v o u) via Gn(v) is introducedin [3], and arose in classifying finiteness and shift dynamics [4]. The earliest example ofcomposite words we are aware of is [56], and [11] is the first of several papers ([55], [31])that discuss composing balanced presentations of the trivial group as it relates to theAndrews- Curtis conjecture. Only [31] specifically considers cyclically presented groups,and they show that if Gn(u) = Gn(v) = 1, then Gn(v o u) = 1. This follows immediatelyfrom our results.
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  • description.provenance : Submitted by Kirk Mc Dermott (mcdermki@oregonstate.edu) on 2017-06-16T20:53:17ZNo. of bitstreams: 2license_rdf: 1223 bytes, checksum: d127a3413712d6c6e962d5d436c463fc (MD5)McDermottKirkM2017.pdf: 3115440 bytes, checksum: 5edb673da41d3ff6336f2f401cb25308 (MD5)
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