arXiv research

Minimal topology on surface homeomorphisms proven.

Study uses equivariant topology to measure distances between G metric spaces.

The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks fa…

Study on hyperconvex representations of hyperbolic groups in complex flag manifolds.

Generic groups can't move spaces but have rich actions.

We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete group…

Study shows conditions for continuity of foliated homeomorphisms action on space of leaves.

New method uses cohomology to quantify molecular similarity.

Let $X$ be a Hausdorff topological group and $G$ a locally compact subgroup of $X$. We show that $X$ admits a locally finite $σ$-discrete $G$-functionally open cover each member of which is $G$-homeomorphic to a twisted product $G\times_H S_i$, where $H$ is a compact large subgroup of $G$ (i.e., the quotient $G/H$ is a…

This paper studies limits of aspherical manifolds with specific curvature conditions.

We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.

The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.

We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a rou…

The paper explores rectifiability in sub-Riemannian geometry, finding a smooth hypersurface with unique properties.

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $Γ$ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit e…

The topological fundamental group $π_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space $X$, we compute the topological fundamental group of the "hoop earring" spac…

We construct a geometric model for the mapping class group M of a non-exceptional oriented surface of finite type and use it to show that the action of M on the compact Hausdorff space of complete geodesic laminations is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every s…

Topology of non-orientable spaces without boundary is studied.

Magnitude is not continuous but may be stable for most finite metric spaces.

Study shows local topologies of certain geometric spaces.

The paper classifies Kleinian groups with Hausdorff dimension less than 1.

The paper connects geometric and topological concepts to bound distances between metric spaces.

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic lamination…

The paper explores conditions for groups of homeomorphisms to be isometrizable.

In this paper, we focus on the geometry of compact conformally flat manifolds $(M^n,g)$ with positive scalar curvature. Schoen-Yau proved that its universal cover $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$ such that $M^n$ is a Kleinian manifold. Moreover, the limit set of the Kleinian group…

Characterizes isolated compact subgroups in Lie groups.

We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved $2$-spheres.

Schwarzschild 3-manifold stability proven for 3D Penrose inequality.

Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F of X such that FG=X and the restriction of the quotient projection X -> X/G to F is a perfect map F -> X/…

Abstract commensurators linked to topological models of solenoids.

Random walks on mapping class groups identified with geodesic laminations.

Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.

This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…

New principles prove precompactness of domains with lower Ricci curvature bound.

The paper studies dynamical properties in semigroups modulo ideals.

In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topolog…

Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.

Measure homology was introduced by Thurston in order to compute the simplicial volume of hyperbolic manifolds. Berlanga endowed measure homology with a structure of graded locally convex (possibly non-Hausdorff) topological vector space. In this note we completely characterize Berlanga's topology on measure homology of…

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds…

We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold $\mathbb{P}^m$ with normal bundle $\oplus_{j=1}^{n-m}\mathcal{O}_…

It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a new topology on the set of fixed endpoint homotopy classes of paths. This version…

We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopted on the Geroch-Kronheime…

Milnor-Thurston homology theory is a construction of homology theory that is based on measures. It is known that it is equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor-Thurst…

This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal posi…

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Lambda(M) be the supremum of the bottom eigenvalue of the Laplacian of N, where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of lim…