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Sergei Buyalo

Publications and source records attributed to Sergei Buyalo.

16 recordsLinked to original sources
arXiv2020

Symmetries of cross-ratios and the equation for Möbius structures

We consider orthogonal representations $η_n:S_n \curvearrowright \mathbb{R}^N$ of the symmetry groups $S_n$, $n\ge 4$, with $N=n!/8$ motivated by symmetries of cross-ratios. For $n=5$ we find the decomposition of $η_5$ into irreducible components and show that one of the components gives the solution to the equations, which describe Möbius structures in the class of sub-Möbius structures. In this sense, the condition defining Möbius structures is hidden already in symmetries of cross-ratios.

math.MG↗
arXiv2019

Inverse problem for Moebius geometry on the circle

We give a solution to the inverse problem of Moebius geometry on the circle. Namely, we describe a class of Moebius structures on the circle for each of which there is a hyperbolic space such that its boundary at infinity is the circle, and the induced Moebius structure coincides with the given one. That class is not empty and form an open neighborhood of the canonical Moebius structure in an appropriate fine topology.

math.MG↗
arXiv2019

SRA-free condition by Zolotov for self-contracted curves and nondegeneracy of zz-distance for Möbius structures on the circle

SRA-free condition for metric spaces (that is, spaces without Small Rough Angles) was introduced by Zolotov to study rectifiability of self-contracted curves in various metric spaces. We give a Moebius invariant version of this notion which allows to show that zz-distance associated with a respective Moebius structure on the circle is nondegenerate. This result is an important part of a solution to the inverse problem of Moebius geometry on the circle.

math.MG↗
arXiv2018

On the inverse problem of Moebius geometry on the circle

Any (boundary continuous) hyperbolic space induces on the boundary at infinity a Moebius structure which reflects most essential asymptotic properties of the space. In this paper, we initiate the study of the inverse problem: describe Moebius structures which are induced by hyperbolic spaces at least in the simplest case of the circle. For a large class of Moebius structures on the circle, we define a canonical "filling" each of them, which serves as a natural candidate for a solution of the inverse problem. This is a 3-dimensional (pseudo)metric space Harm, which consists of harmonic 4-tuples of the respective Moebius structure with a distance determined by zig-zag paths. Our main result is the proof that every line in Harm is a geodesic, i.e., shortest in the zig-zag distance on each segment. This gives a good starting point to show that Harm is Gromov hyperbolic with the prescribed Moebius structure at infinity.

math.MG↗
arXiv2017

Spectral geometries on a compact metric space

The notion of a spectral geometry on a compact metric space X is introduced. This notion serves as a discrete approximation of X motivated by the notion of a spectral triple from non-commutative geometry. A set of axioms charaterising spectral geometries is given. Bounded deformations of spectral geometries are studied and the relationship between the dimension of a spectral geometry and more traditional dimensions of metric spaces is investigated.

math.OA↗
arXiv2017

Möbius structures and timed causal spaces on the circle

We discuss a conjectural duality between hyperbolic spaces on one hand and spacetimes on the other hand, living on the opposite sides of the common absolute. This duality goes via Möbius structures on the absolute, and it is easily recognized in the classical case of symmetric rank one spaces. In a general case, no trace of such duality is known. As a first step in this direction, we show how Möbius structures on the circle from a large class including those which stem from hyperbolic spaces give rise to 2-dimensional spacetimes, which are axiomatic versions of de Sitter 2-space, and vice versa. The paper has two Appendices, one of which is written by V.Schroeder.

math.MG↗
arXiv2016

Moebius and sub-Moebius structures

We introduce a notion of a sub-Moebius structure and find necessary and sufficient conditions under which a sub-Moebius structure is a Moebius structure. We show that on the boundary at infinity of every Gromov hyperbolic space Y there is a canonical sub-Moebius structure which is invariant under isometries of Y such that the sub-Moebius topology on the boundary coincides with the standard one.

math.MG↗
arXiv2012

Moebius characterization of the boundary at infinity of rank one symmetric spaces

A Moebius structure (on a set X) is a class of metrics having the same cross-ratios. A Moebius structure is ptolemaic if it is invariant under inversion operations. The boundary at infinity of a CAT(-1) space is in a natural way a Moebius space, which is ptolemaic. We give a free of classification proof of the following result that characterizes the rank one symmetric spaces of noncompact type purely in terms of their Moebius geometry: Let X be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then X is Moebius equivalent to the boundary at infinity of a rank one symmetric space.

math.MG↗
arXiv2010

Moebius structures and Ptolemy spaces: boundary at infinity of complex hyperbolic spaces

The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric space of noncompact type. We prove this conjecture for the class of complex hyperbolic spaces as our main result.

math.MG↗
arXiv2009

Boundary at infinity of symmetric rank one spaces

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in the space and on the boundary at infinity respectively.

math.DG↗
arXiv2005

Dimension of locally and asymptotically self-similar spaces

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the topological dimension of the subspace. As an application of the first result, we prove the Gromov conjecture that the asymptotic dimension of every hyperbolic group G equals the topological dimension of its boundary at infinity plus 1, asdim G=dim(dG)+1. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension, in particular, those are first examples of metric spaces X, Y with asdim(X x Y)<asdim X+asdim Y. Other applications are also given.

math.GT↗
arXiv2003

Metrics of nonpositive curvature on graph-manifolds and electromagnetic fields on graphs

A 3-dimensional graph-manifold is composed from simple blocks which are products of compact surfaces with boundary by the circle. Its global structure may be as complicated as one likes and is described by a graph which might be an arbitrary graph. A metric of nonpositive curvature on such a manifold, if it exists, can be described essentially by a finite number of parameters which satisfy a geometrization equation. The aim of the work is to show that this equation is a discrete version of the Maxwell equations of classical electrodynamics, and its solutions, i.e., metrics of nonpositive curvature, are critical configurations of the same sort of action which describes the interaction of an electromagnetic field with a scalar charged field. We establish this analogy in the framework of the spectral calculus (noncommutative geometry) of A. Connes.

math-ph↗
arXiv2001

Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant $\subcorank X$ of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for each $t\in T$ the set $g^{-1}(t)$ has subexponential growth rate in $X$ and the topological dimension $\dim T=k$ is minimal among all such maps. Our main result is the inequality $\hyprank X\le\subcorank X$ for a large class of metric spaces $X$ including all locally compact Hadamard spaces, where $\hyprank X$ is maximal topological dimension of $\di Y$ among all $\CAT(-1)$ spaces $Y$ quasi-isometrically embedded into $X$ (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of $\hyprank$ conjectured by M. Gromov, in particular, that any Riemannian symmetric space $X$ of noncompact type possesses no quasi-isometric embedding $\hyp^n\to X$ of the standard hyperbolic space $\hyp^n$ with $n-1>\dim X-\rank X$.

math.DG↗