arXiv subjects

Alina Vdovina

Publications and source records attributed to Alina Vdovina.

43 records · Page 3Linked to original sources

arXiv2006

Noncommutative geometry on trees and buildings

We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization of the algebra of the dual graph of the special fiber of the Mumford curve and a variant of the Antonescu-Christensen spectral geometries for AF algebras. The information on the Schottky uniformization is encoded in the spectral geometry through the Patterson-Sullivan measure on the limit set. Some higher rank cases are obtained by adapting the construction for trees.

math.QA↗

arXiv2005

Triangular hyperbolic buildings

We construct triangular hyperbolic polyhedra whose links are generalized 4-gons. The universal cover of those polyhedra are hyperbolic buildings, which appartments are hyperbolic planes tesselated by regular triangles with angles $π/4$. Moreover, the fundamental groups of the polyhedra acts simply transitively on vertices of the buildings.

math.CO↗

arXiv2004

$C^*$-algebras coming from some buildings

We construct compact polyhedra with triangular faces whose links are generalized 3-gons. They are interesting compact spaces covered by Euclidean buildings of type $A_2$. Those spaces give us two-dimensional subshifts, which can be used to construct some $C^*$-algebras.

math.CO↗

arXiv2003

On the number of extremal surfaces

Let $X$ be a compact Riemann surface of genus $\geq 2$ of constant negative curvature -1. An extremal disk is an embedded (resp. covering) disk of maximal (resp. minimal) radius. A surface containing an extremal disk is an {\em extremal surface}. This paper gives formulas enumerating extremal surfaces of genus $\geq 4$ up to isometry. We show also that the isometry group of an extremal surface is always cyclic of order 1, 2, 3 or 6.

math.DG↗

arXiv2003

Groups, periodic planes and hyperbolic buildings

We give an elementary construction of polyhedra whose links are connected bipartite graphs, which are not necessarily isomorphic pairwise. We show, that the fundamental groups of some of our polyhedra contain surface groups. In particular, we construct polyhedra whose links are generalized $m$-gons. The polyhedra of this type are interesting because of their universal coverings, which are two-dimensional hyperbolic buildings with different links. The presentation of the results is done in the language of combinatorial group theory.

math.CO↗

arXiv2003

Polyhedra with specified links

We construct compact polyhedra with $m$-gonal faces whose links are generalized 3-gons. It gives examples of cocompact hyperbolic bildings of type $P(m,3)$. For $m=3$ we get compact spaces covered by Euclidean buildings of type $A_2$.

math.CO↗

arXiv2001

Counting 1-vertex Triangulations Of Oriented Surfaces

A {\em $1-$vertex triangulation} of an oriented compact surface $S$ of genus $g$ is an embedded graph $T\subset S$ with a unique vertex such that all connected components of $S\setminus T$ are triangles (adjacent to exactly 3 edges of $T$). This paper gives formulas enumerating such triangulations (up to equivalence) on an oriented surface of given genus. {\em Une triangulation à un sommet} d'une surface orientée compacte $S$ de genre $g$ est un graphe $T\subset S$ qui a un unique sommet et dont toutes les faces (composantes connexes de $S\setminus T$) sont des triangles (incidentes à trois arêtes de $T$). Cet article donne des formules permettant d'énumérer ces triangulations.

math.CO↗