arXiv subjects

Boris N. Apanasov

Publications and source records attributed to Boris N. Apanasov.

4 recordsLinked to original sources

arXiv2024

Rigidity of locally symmetric rank one manifolds of infinite volume

We discuss questions by Mostow \cite{Mo1}, Bers \cite{B} and Krushkal \cite{Kr1, Kr2} about uniqueness of a conformal or spherical CR structure on the sphere at infinity $\partial H_\mathbb{F}^n$ of symmetric rank one space $H_\mathbb{F}^n$ over division algebra $\mathbb{F}=\mathbb{R}\,,\mathbb{C}\,,\mathbb{H}\,,\text{or}\,\, \mathbb{O} $ compatible with the action of a discrete group $G\subset\operatorname{Isom}H_\mathbb{F}^n$. Introducing a nilpotent Sierpiński carpet with a positive Lebesgue measure in the nilpotent geometry in $\partial H_\mathbb{F}^n\setminus\{\infty\}$ and its stretching, we construct a non-rigid discrete $\mathbb{F}$-hyperbolic groups $G\subset\operatorname{Isom}H_\mathbb{F}^n$ whose non-trivial deformations are induced by $G$-equivariant homeomorphisms of the space. Here we consider two situations: either the limit set $Λ(G)$ is the whole sphere at infinity $\partial H_\mathbb{F}^n$ or restrictions of such non-trivial deformations to components of the discontinuity set $Ω(G)\subset \partial H_\mathbb{F}^n$ are given by restrictions of $\mathbb{F}$-hyperbolic isometries. In both cases the demonstrated non-rigidity is related to non-ergodic dynamics of the discrete group action on the limit set which could be the whole sphere at infinity.

math.GT↗

arXiv2019

Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball $B^3\subset \mathbb{R}^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices $Γ\subset \operatorname{Isom} H^3$ in the unit 3-ball and with its discrete representation $G=ρ(Γ)\subset \operatorname{Isom} H^4 $. Here $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cupΩ(G))/G$ and the kernel of the homomorphism $ρ\!:\! Γ\rightarrow G$ is a free group $F_3$ on three generators.

math.GT↗

arXiv2017

Topological barriers for locally homeomorphic quasiregular mappings in 3-space

We construct a new type of locally homeomorphic quasiregular mappings in the 3-sphere and discuss their relation to the M.A.Lavrentiev problem, the Zorich map with an essential singularity at infinity, the Fatou's problem and a quasiregular analogue of domains of holomorphy in complex analysis. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such locally homeomorphic quasiregular mappings are defined in the 3-sphere $S^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic 3-lattices $Γ$ in the unit 3-ball and its complement in $S^3$ and with its discrete representation $G=ρ(Γ)$ in the group of isometries of $H^4 $. Here $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cupΩ(G))/G$ and the kernel of the homomorphism $ρ\!:\! Γ\rightarrow G$ is a free group $F_3$ on three generators.

math.CV↗

arXiv2016

Group actions, Teichmüller spaces and cobordisms

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary $\partial M$. In addition to the standard conformal ergodic action of a uniform hyperbolic lattice on the round sphere $S^{n-1}$ and its quasiconformal deformations in $S^n$, we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed $n$-ball into $S^n$), on non-trivial $(n-1)$-knots in $S^{n+1}$, as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from "non-standard" components of its varieties of representations (faithful or with large kernels of defining homomorphisms).

math.GT↗