Margarita Amchislavska

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 $\mathbb{Z} \wr \mathbb{Z}$), 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 $\Gamma_2=\langle a,s, t~|~ [a,a^t]=1, \ [s,t]=1, \ a^s=aa^t\rangle$. 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.