MiS Preprint Repository

We study the geometry and topology of generalized lattices in topological vector spaces, and their quotients. A subgroup G of a topological vector space V is a generalized lattice if G is line-free and exhaustive in the sense that the closure of its linear span in V is V. We first discuss stronger conditions than line-free and exhaustive, and give examples of generalized lattices. We show that the sum of a locally generated, exhaustive subgroup G and a cocompact subgroup H in a topological vector space V is dense in V. We next study the groups $G^P:=L^P([0,1],\mathbb{Z})\subset L^P:=([0,1],\mathbb{R})$ for $1\le p\le \mathrm{\infty }$ mainly from the standpoint of geometry. We define geometrically significant global and local contractions of $G^P$ and $L^P/G^P$, and prove self-similarity properties of $G^P$. We next study the rectifiability and differentiability properties of curves in Banach spaces and $G^P$, especially the singular nature of curves in $G^P$. Following this we study when the (generalized) universal cover of a quotient V/G is again a topological vector space. Finally, we prove the existence of a (possibly non-convex) Dirichlet-Voronoi fundamental set of G in V, when V is a uniformly strongly convex Banach space and V/G is geodesic space. We also show this existence of such a fundamental set for the subgroup $G^P$ of $L^P$. We conclude with unsolved questions.