arXiv research

Study mapping class group actions on surface fundamental groups.

We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow f…

Study virtual fundamental classes of derived manifolds, proving invariant vanishes.

Continues work on derived manifolds and symplectic schemes, constructing virtual classes.

We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the relative group homology and quandle homology from the viewpoints of Inoue--Kabay…

Confirming a conjecture, we show fundamental groups of certain abelian differentials are framed mapping class groups.

We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of $M$ vanishes if $M$ is geometrically finite. Furthermore, when $M$ is a $\mathbb{R}$-rank one locally symmetric space, we show that the bou…

Study finds bounds for fundamental tone on special Riemannian manifolds.

In this paper we show that the fibered isomorphism conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the neg…

Study of fundamental groups of 3D small covers using Morse theory.

Explicit presentations found for asymptotically rigid mapping class groups.

The theorem connects surface mapping groups to fundamental groupoids.

New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.

New algebraic fundamental groups identified for fake projective planes.

Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.

Method constructs fundamental domains for Picard modular groups.

Let $Σ$ be a surface whose interior admits a hyperbolic structure of finite volume. In this paper, we show that any infinite order mapping class acts with infinite order on the homology of some universal $k$--step nilpotent cover of $Σ$. We show that a Torelli mapping class either acts with infinite order on the homolo…

Our goal is to better understand the relationship between the polyhedron and the group associated with a fundamental domain in H^3. In this paper, we will study torsion-free groups and determine a formula for how many edge classes a given abstract polyhedron must have. We will use that result to classify all fundamenta…

Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.

In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fun…

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M can be expressed via these new integrals. The ring of exponential iterated inte…

Study shows mapping class group action is ergodic on specific representations.

In this paper we study some consequences of the author's classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by …

We find the fundamental solution to the p-Laplace equation in a class of Hörmander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space…

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of 3-manifolds to SL(4,R) that factor over SL(2,C), in particular for those factoring over the isomorphism PSL(2,C) = S0(3,1). We also discuss consequences for the number of connected components of SL(4,R)-character…

This work examines fundamental limits in model falsification without assuming specific distributions.

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it i…

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented t…

We study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts: - Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes of generating pairs. - For any natural number N there is a closed hyperbolic 3-manifold whose fundamental group has N d…

We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…

We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperb…

A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of the closure o…

New findings show fundamental group is not audible in spherical space forms.

Algorithm computes fundamental classes of spin components in moduli space.

We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinat…

Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension $3$ (PD$_{3}$-complexes) generalize to PD$_{n}$-complexes with $(n-2)$-connected universal cover for $n \ge 3$. Baues and Bleile showed that such complexes are classified, up to oriented homotopy eq…

Homotopy classification for certain 4-manifolds with dihedral fundamental groups.

Homotopy types of 4-manifolds tied to their fundamental groups.

We show that the resulting manifold by $r$-surgery on a large class of two-bridge knots has left-orderable fundamental group if the slope $r$ satisfies certain conditions. This result gives a supporting evidence to a conjecture of Boyer, Gordon and Watson that relates $L$-spaces and the left-orderability of their funda…

Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective ma…

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…

Study Kähler groups from orbifold compactifications of curve moduli.

In this paper, we study the fundamental group of a certain class of globally hyperbolic Lorentzian manifolds with a positive curvature tensor. We prove that the fundamental group of lightlike geodesically complete parametrized Lorentzian products is finite under the conditions of a positive curvature tensor and the fib…

Given any connected compact orientable surface, a pair of mapping classes are said to be procongruently conjugate if they induce a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every char…

By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.

In this work we will build a fundamental domain for Deligne-Mostow lattices in PU(2,1) with 2-fold symmetry, which complete the whole list of Deligne-Mostow lattices in dimension 2. These lattices were introduced by Deligne and Mostow using monodromy of hypergeometric functions and have been reinterpreted by Thurston a…

Study of Sasaki groups vs Kähler groups, showing distinct behaviors.

Study handles in 4-manifolds with cyclic fundamental group.