Related papers: A class of Latt\`es maps with cellular structures
This paper is concerned with realizing Lattes maps as subdivision maps of finite subdivision rules. The main result is that the Lattes maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite…
In this paper, we consider boundary extensions of two classes of mappings between metric measure spaces. These two mapping classes extend in particular the well-studied geometric mappings such as quasiregular mappings with integrable…
We define a new quasiregular mapping T in three dimensions that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family \lambda T for \lambda>0,…
In this paper, left-invariant almost contact metric structures on three-dimensional non-unimodular Lie groups are investigated. It is proved that for every Riemannian Lie group, there is one of these structures. In addition, left-invariant…
We establish a rescaling theorem for isolated essential singularities of quasiregular mappings. As a consequence we show that the class of closed manifolds receiving a quasiregular mapping from a punctured unit ball with an essential…
This article is the introductory part of authors PhD thesis. The article presents a new coordinate invariant definition of quasiregular and quasiconformal mappings on Riemannian manifolds that generalizes the definition of quasiregular…
We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them…
For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on…
We introduce a class of graphs with coloured edges to encode subsystems of the classical root systems, which in particular classify them up to equivalence. We further use the graphs to describe root-kernel intersections, as well as…
The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, projective plane and the…
We consider the lattice of coarse structures on a set $X$ and study metrizable, locally finite and cellular coarse structures on $X$ from the lattice point of view.
This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ${\mathbb P}^n$ ({\it quasi-lines}). Various characterizations of a Zariski neighbourhood of a…
A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$\kappa$-to-one in a natural sense for some $\kappa>0$. For non-amenable graphs, all quasi-isometries are quasi-$\kappa$-to-one for any…
We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…
We introduce asymptotic-M\"obius (AM) maps, a large-scale analogue of quasi-M\"obius maps tailored to geometric group theory. AM-maps capture coarse cross-ratio behavior for configurations of points that lie far apart, providing a notion of…
We characterize all translation invariant half planar maps satisfying a certain natural domain Markov property. For p-angulations with p \ge 3 where all faces are simple, we show that these form a one-parameter family of measures…
We show that finitely presented groups which admit $k$-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, we answer a question of Georgakopoulos and Papasoglu in the special case of coarsely…
There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets are quasimodular forms. We extend these families so that the corresponding q-brackets are quasimodular for a…
We study the equivariant cohomology of the moduli space of quasimaps from $\mathbb{P}^1$ with one marked point to the flag variety. This moduli space has an open subset isomorphic to the Laumon space. The equivariant cohomology of the…
We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single…