Searcharxiv⌕ Search

arXiv subjects

Anna Pratoussevitch

Publications and source records attributed to Anna Pratoussevitch.

16 recordsLinked to original sources
arXiv2025

Stressability of Semi-Discrete Frameworks

In this paper we study the stressability of semi-discrete frameworks in the plane which are generated by a discrete sequence of smooth curves. We characterize their stressability property by the existence of stresses fulfilling certain difference-differential equation. In particular, we define a semi-discrete height function which we use to generate liftings. Furthermore, we show a semi-discrete analogue of the Maxwell-Cremona lifting property which implies that the stressable semi-discrete frameworks in the plane are precisely the orthogonal projections of semi-discrete conjugate surfaces in 3-space. Finally, we discuss geometric implications for frameworks with vanishing boundary forces and characterize the liftability of frameworks which only consist of two neighboring curves forming one strip.

math.MG↗
arXiv2024

Circumscribed Circles in Integer Geometry

Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles in integer geometry. We introduce the notions of integer and rational circumscribed circles of integer sets. We determine the conditions for a finite integer set to admit an integer circumscribed circle and describe the spectra of radii for integer and rational circumscribed circles.

math.NT↗
arXiv2024

Farey Bryophylla

The construction of the Farey tessellation in the hyperbolic plane starts with a finitely generated group of symmetries of an ideal triangle, i.e. a triangle with all vertices on the boundary. It induces a remarkable fractal structure on the boundary of the hyperbolic plane, encoding every element by the continued fraction related to the structure of the tessellation. The problem of finding a generalisation of this construction to the higher dimensional hyperbolic spaces has remained open for many years. In this paper we make the first steps towards a generalisation in the three-dimensional case. We introduce conformal bryophylla, a class of subsets of the boundary of the hyperbolic 3-space which possess fractal properties similar to the Farey tessellation. We classify all conformal bryophylla and study the properties of their limiting sets.

math.GT↗
arXiv2024

Complex Hyperbolic Triangle Groups of Type $[m,m,0; n_1, n_2, 2]$

In this paper we study the discreteness of complex hyperbolic triangle groups of type $[m,m,0; n_1, n_2, 2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $n_1, n_2, 2$ in complex geodesics with pairwise distances $m, m, 0$. For fixed $m$, the parameter space of such groups is of real dimension one. We determine the possible orders for $n_1$ and $n_2$ and also intervals in the parameter space that correspond to discrete and non-discrete triangle groups.

math.GT↗
arXiv2020

Hyperbolic Groups and Non-Compact Real Algebraic Curves

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs $(P,τ)$, where $P$ is a compact Riemann surface with a finite number of holes and punctures and $τ:P\to P$ is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.

math.AG↗
arXiv2020

Complex Hyperbolic Triangle Groups of Type $[m,m,0;3,3,2]$

In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $3,3,2$ in complex geodesics with pairwise distances $m,m,0$. For fixed $m,$ the parameter space of such groups is of real dimension one. We determine intervals in this parameter space that correspond to discrete and to non-discrete triangle groups.

math.GT↗
arXiv2019

Two series of polyhedral fundamental domains for Lorentz bi-quotients

The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of the form $Γ_1\backslash G/Γ_2$, where $G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(2,{\mathbb R})}$ is a simply connected Lie group with the Lorentz metric given by the Killing form, $Γ_1$ and $Γ_2$ are discrete subgroups of $G$ and $Γ_2$ is cyclic. A construction of polyhedral fundamental domains for the action of $Γ_1\timesΓ_2$ on $G$ via $(g,h)\cdot x=gxh^{-1}$ was given in the earlier work of the second author. In this paper we give an explicit description of the fundamental domains obtained by this construction for two infinite series of groups. These results are connected to singularity theory as the bi-quotients $Γ_1\backslash G/Γ_2$ appear as links of certain quasi-homogeneous $\mathbb Q$-Gorenstein surface singularities, i.e.\ the intersections of the singular variety with sufficiently small spheres around the isolated singular point.

math.DG↗
arXiv2019

Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type $[m_1,m_2,0]$

In this paper we consider ultra-parallel complex hyperbolic triangle groups of type $[m_1,m_2,0]$, i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness results for these groups and discuss the connection between the discreteness results and ellipticity of certain group elements.

math.GT↗
arXiv2016

Higher Spin Klein Surfaces

We find all m-spin structures on Klein surfaces of genus larger than one. An m-spin structure on a Riemann surface P is a complex line bundle on P whose m-th tensor power is the cotangent bundle of P. A Klein surface can be described by a pair (P,tau), where P is a Riemann surface and tau is an anti-holomorphic involution on P. An m-spin structure on a Klein surface (P,tau) is an m-spin structure on the Riemann surface P which is preserved under the action of the anti-holomorphic involution tau. We determine the conditions for the existence and give a complete description of all real m-spin structures on a Klein surface. In particular, we compute the number of m-spin structures on a Klein surface (P,tau) in terms of its natural topological invariants.

math.AG↗
arXiv2016

Moduli Spaces of Higher Spin Klein Surfaces

We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin bundle on a Klein surface is a complex line bundle whose m-th tensor power is the cotangent bundle. The spaces of higher spin bundles on Klein surfaces are important because of their applications in singularity theory and real algebraic geometry, in particular for the study of real forms of Gorenstein quasi-homogeneous surface singularities. In this paper we describe all connected components of the space of higher spin bundles on hyperbolic Klein surfaces in terms of their topological invariants and prove that any connected component is homeomorphic to a quotient of a Euclidean space by a discrete group. We also discuss applications to real forms of Brieskorn-Pham singularities.

math.AG↗
arXiv2010

The Combinatorial Geometry of Q-Gorenstein Quasi-Homogeneous Surface Singularities

The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie group G~ gives rise to a bi-invariant Lorentz metric of constant curvature. We consider a discrete subgroup Gamma_1 and a cyclic discrete subgroup Gamma_2 in G~ which satisfy certain conditions. We describe the Lorentz space form Gamma_1\G~/Gamma_2 by constructing a fundamental domain for the action of the product of Gamma_1 and Gamma_2 on G~ by (g,h)*x=gxh^{-1}. This fundamental domain is a polyhedron in the Lorentz manifold G~ with totally geodesic faces. For a co-compact subgroup the corresponding fundamental domain is compact. The class of subgroups for which we construct fundamental domains corresponds to an interesting class of singularities. The bi-quotients of the form Gamma_1\G~/Gamma_2 are diffeomorphic to the links of quasi-homogeneous Q-Gorenstein surface singularities.

math.DG↗
arXiv2010

Non-Discrete Complex Hyperbolic Triangle Groups of Type (m,m,infinity)

In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.

math.DG↗
arXiv2009

Higher Arf Functions and Topology of the Moduli Space of Higher Spin Riemann Surfaces

We prove that any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action. Our proof is based on the representation of the space of m-spin structures on a Riemann surface as a finite affine space of Z/mZ-valued functions on the fundamental group of the surface.

math.AG↗
arXiv2004

Fundamental Domains in Lorentzian Geometry

We consider discrete subgroups Gamma of the simply connected Lie group SU~(1,1), the universal cover of SU(1,1), of finite level, i.e. the subgroup intersects the centre of SU~(1,1) in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curvature on the Lie group SU~(1,1). The discrete subgroup Gamma acts on SU~(1,1) by left translations. We describe the Lorentz space form SU~(1,1)/Gamma by constructing a fundamental domain F for Gamma. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Gamma satisfying the following condition: The image of Gamma in PSU(1,1) has a fixed point u in the unit disk of order larger than the index of Gamma. The construction depends on the group Gamma and on the orbit Gamma(u) of the fixed point u.

math.DG↗
arXiv2004

Traces in Complex Hyperbolic Triangle Groups

We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant alpha of triangles in the complex hyperbolic plane. The main result of the paper is a formula, which expresses the trace of an element of the group as a Laurent polynomial in exp(i alpha) with coefficients independent of alpha and computable using a certain combinatorial winding number. We also give a recursion formula for these Laurent polynomials and generalise the trace formulas for the groups generated by complex mu-reflections. We apply these formulas to prove some discreteness and some non-discreteness results for complex hyperbolic triangle groups.

math.DG↗