A note on lattice knots
The aim of this note is to share the observation that the set of elementary operations of Turing on lattice knots can be reduced to just one type of simple local switches.
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arXiv subjects
Sasha Anan'in
Publications and source records attributed to Sasha Anan'in.
Moduli spaces of polygons and deformations of polyhedra with boundary
We prove a conjecture of Ian Agol: all isometric realizations of a polyhedral surface with boundary sweep out an isotropic subset in the Kapovich-Millson moduli space of polygons isomorphic to the boundary. For a generic polyhedral disk we show that boundaries of its isometric realizations make up a Lagrangian subset. As an application of this result, we obtain a new solution to the problem of Richard Kenyon about spanning domes of piecewise linear curves comprised of unit intervals in R^3.
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A couple of real hyperbolic disc bundles over surfaces
Applying the techniques developed in [AGG], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [GLT], such bundles $M\to S$ should satisfy the inequality $|eM/χS|\leqslant1$, where $eM$ stands for the Euler number of the bundle and $χS$, for the Euler characteristic of the surface. In this paper, we construct new examples that provide a maximal value of $|eM/χS|=\frac35$ among all known examples. The former maximum, belonging to Feng Luo [Luo], was $|eM/χS|=\frac12$.
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Hyperbolic 2-spheres with cone singularities
We study the space $C(a_0,a_1,\dots,a_n)$ of hyperbolic 2-spheres with cone points of prescribed apex curvatures $2a_0,2a_1,\dots,2a_n\in]0,2π[$ and some related spaces. For $n=3$, we get a detailed description of such spaces. The euclidean 2-spheres were considered by W. P. Thurston: for $n=4$, the corresponding spaces provide the famous 7 examples of nonarithmetic compact holomorphic 2-ball quotients previously constructed by Deligne-Mostow.
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Grassmannians and conformal structure on absolutes
We study grassmannians associated with a linear space with a nondegenerate hermitian form. The geometry of these grassmannians allows us to explain the relation between a (pseudo-)riemannian projective geometry and the conformal structure on its ideal boundary (absolute). Such relation encompasses, for instance, the usual conformal structure on the absolute of real hyperbolic space, the usual conformal structure on the absolute of de Sitter space, the conformal contact structure on the absolute of complex hyperbolic space, and the causal structure on the absolute of anti-de Sitter space.
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Complex hyperbolic equidistant loci
We describe and study the loci equidistant from finitely many points in the so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic $2$-ball $\Bbb B$. In particular, we show that the bisectors (= the loci equidistant from $2$ points) containing the (smooth real algebraic) curve equidistant from given $4$ generic points form a real elliptic curve and that the foci of the mentioned bisectors constitute an isomorphic elliptic curve. We are going to use the obtained facts in constructions of (compact) quotients of $\Bbb B$ by discrete groups. With similar technique, we also classify up to isotopy generic $3$-dimensional algebras (i.e., bilinear operations) over an algebraically closed field $\Bbb K$ of characteristic $\ne2,3$. Briefly speaking, an algebra is classified by the (plane projective) curve $D$ of its zero divisors equipped with a nonprojective automorphism of $D$. This classification is almost equivalent to the classification of the so-called geometric tensors given in [BoP] by A. Bondal and A. Polishchuk in their study of noncummutative projective planes.
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Poincaré's polyhedron theorem for cocompact groups in dimension 4
We prove a version of Poincaré's polyhedron theorem whose requirements are as local as possible. New techniques such as the use of discrete groupoids of isometries are introduced. The theorem may have a wide range of applications and can be generalized to the case of higher dimension and other geometric structures. It is planned as a first step in a program of constructing compact $\mathbb C$-surfaces of general type satisfying $c_1^2=3c_2$.
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Any component of moduli of polarized hyperkaehler manifolds is dense in its deformation space
Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.
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Reflections, bendings, and pentagons
We study relations between reflections in (positive or negative) points in the complex hyperbolic plane. It is easy to see that the reflections in the points q_1,q_2 obtained from p_1,p_2 by moving p_1,p_2 along the geodesic generated by p_1,p_2 and keeping the (dis)tance between p_1,p_2 satisfy the bending relation R(q_2)R(q_1)=R(p_2)R(p_1). We show that a generic isometry F\in SU(2,1) is a product of 3 reflections, F=R(p_3)R(p_2)R(p_1), and describe all such decompositions: two decompositions are connected by finitely many bendings involving p_1,p_2/p_2,p_3 and geometrically equal decompositions differ by an isometry centralizing F. Any relation between reflections gives rise to a representation H_n->PU(2,1) of the hyperelliptic group H_n generated by r_1,...,r_n with the defining relations r_n...r_1=1, r_j^2=1. The theorem mentioned above is essential to the study of the Teichmuller space TH_n. We describe all nontrivial representations of H_5, called pentagons, and conjecture that they are faithful and discrete.
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Yet Another Poincare's Polyhedron Theorem
Poincaré's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincaré's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of nonconstant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice.
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Coordinate-free classic geometries
This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features of the approach. The framework introduced here has already been shown to be adequate for solving problems concerning particular classic spaces.
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Complex Hyperbolic Structures on Disc Bundles over Surfaces
We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->Σ that: admit both real and complex hyperbolic structures; satisfy the equality 2(χ+e)=3τ; satisfy the inequality χ/2 PU(2,1) with fractional Toledo invariant; where χ is the Euler characteristic of Σ, e denotes the Euler number of M, and τ stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(χ+e)=3τ, we conjecture that there exists a holomorphic section in all our examples. In order to reduce the amount of calculations, we systematically explore coordinate-free methods.
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On `A characterization of R-Fuchsian groups acting on the complex hyperbolic plane'
We indicate a C-Fuchsian counter-example to the result with the above title announced at http://www.maths.dur.ac.uk/events/Meetings/LMS/2011/GAL11/program.pdf and prove a stronger statement.
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Basic coordinate-free non-Euclidean geometry
These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.
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Differential geometry of grassmannians and Plucker map
Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For `hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the `elliptic' case.
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A Hyperelliptic View on Teichmuller Space. I
We explicitly describe the Teichmuller space TH_n of hyperelliptic surfaces in terms of natural and effective coordinates as the space of certain (2n-6)-tuples of distinct points on the ideal boundary of the Poincare disc. We essentially use the concept of a simple earthquake which is a particular case of a Fenchel-Nielsen twist deformation. Such earthquakes generate a group that acts transitively on TH_n. This fact can be interpreted as a continuous analog of the well-known Dehn theorem saying that the mapping class group is generated by Dehn twists. We find a simple and effective criterion that verifies if a given representation of the surface group π_1Σin the group of isometries of the hyperbolic plane is faithful and discrete. The article also contains simple and elementary proofs of several known results, for instance, of W. M. Goldman's theorem [Gol1] characterizing the faithful discrete representations as having maximal Toledo invariant (which is essentially the area of the representation in the two-dimensional case).
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A Hyperelliptic View on Teichmuller Space. II
Using the methods of the previous paper [ABG], we show that the Teichmuller space T of all closed Riemann surfaces is fibred twice over the Teichmuller space H of hyperelliptic ones. Both fibre bundles π_1,π_2:T->H are real algebraic (rational). They define an embedding T->HxH. In addition, we indicate slight modifications of the proof of [ABG, Theorem 5.1] providing an elementary proof of Toledo's rigidity theorem.
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Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle
This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface $Σ$ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open Question 8.1] if a trivial circle bundle over a closed surface of genus >1 admits a holomorphically fillable contact structure. The constructed example M satisfies the relation $2(χ+e)=3τ$ which is necessary for the existence of a holomorphic section of the bundle, where $χ=χΣ$ stands for the Euler characteristic of $Σ$, e=eM, for the Euler number of the bundle, and $τ$, for the Toledo invariant. (The relation is also valid for the series of examples constructed in [AGG].) Open question: Does there exist a holomorphic section of the bundle M?
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