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Results on Goeritz Groups and Farey Trisections
A Heegaard splitting is a decomposition of a 3−manifold, M, into two3−dimensional 1−handlebodies. These 1−handlebodies meet along the splitting surface Σ. The Goeritz group of a splitting is the group of maps which are ambient isotopic to the identity on M, but which fix Σ setwise. In this thesis, we approach the problem of finitely generating the genus g + 1 Goeritz group of #gS1×S2, G(g+1, g), from a couple of different angles. As partial progress toward the resolution of the finite generation problem, we derive several technical results which are analogous to key results in comparable settings. First, we define a generating set which we believe finitely generates G(g +1, g). This generating set was used by both Martin Scharlemann and Jesse Johnson. We define a 1−dimensional simplicial complex Γg, called the dual disk complex, which consists of pairs of disks in the Heegaard splitting which intersect once. We prove that the dual disk complex is connected if and onlyif G(g +1, g) is finitely generated. From there, we develop arguments towardsan inductive proof that Γg is connected; Cho and Koda have established the base case Γ1.Attempting the same problem from a different approach, we also define the special sphere complex, Sg, which consists of the stabilizing curves in the splitting. We show that Sg is connected if and only if Γg is connected. Weobtain a technical result concerning special spheres that are not connected to each other in Sg; an almost identical technical result is used in Goeritz’ original proof. We conclude our efforts with proof that special spheres intersecting at most twelve times are connected in Sg. A trisection of a 4−manifold X4 is a decomposition of a 4−manifold into three 4−dimensional 1−handlebodies with some additional restraints. Trisections were introduced by David Gay and Rob Kirby in 2011 and are a novel way to study 4−manifolds. Low genus trisections are of great interest. Genustwo trisections are known to be standard. In 2019, at the UGA Spring Trisectors meeting, a combinatorially constructed class of diagrams was introduced, called the Farey diagrams. At the time, except for when the Farey diagram was a spun lens space, not much was known about the Farey diagrams or what manifolds they trisect. We present joint work between the author and Román Aranda which establishes that the Farey trisection diagrams, in most cases, are standard.
Moeller, Jesse, "Results on Goeritz Groups and Farey Trisections" (2021). ETD collection for University of Nebraska - Lincoln. AAI28864971.