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The Geometry Of Generalized Lamplighter Groups

Author
Amchislavska, Margarita
Abstract
This work examines geometric properties of generalized lamplighter groups. The thesis contains two parts. The first part gives an elementary account of Bartholdi, Neuhauser and Woess's result that the Cayley graphs of a family of metabelian groups can be realized as 1-skeleta of horocyclic products of trees, extends the result to a wider family of groups (including an infinite valence case, like Z ≀ Z), and makes the translation between the algebraic and geometric descriptions explicit. The second part examines important geometric properties of Baumslag and Remeslennikov's metabelian group [GAMMA]2 = a, s, t | [a, at ] = 1, [s, t] = 1, a s = aat . We show that the Cayley 2-complex of a suitable presentation of [GAMMA]2 is a horocyclic product of three infinitely branching trees. We prove that the subgroup generated by a is undistorted in [GAMMA]2 . Finally, we reduce the question of finding an upper bound on the filling length function of [GAMMA]2 to a combinatorial question about propagating configurations on a two-dimensional rhombic grid.
Date Issued
2014-08-18Subject
lamplighter groups; subgroup distortion; filling length function
Committee Chair
Riley, Timothy R.
Committee Member
Vogtmann, Karen L; Hatcher, Allen E
Degree Discipline
Mathematics
Degree Name
Ph. D., Mathematics
Degree Level
Doctor of Philosophy
Type
dissertation or thesis