arXiv research

Proves Congruence Subgroup Property for two types of groups.

Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian ge…

This paper is a study of the subgroups of the mapping class groups of Riemann surfaces, called "geometric" subgroups, corresponding to the inclusion of subsurfaces. Our analysis includes surfaces with boundary and with punctures. The centres of all the mapping class groups are calculated. We determine the kernel of inc…

We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Joh…

The outer automorphism group Out(F_2g) of a free group on 2g generators naturally contains the mapping class group of a punctured surface as a subgroup. We define a subsurface projection of the sphere complex of the connected sum of n copies of S^1 x S^2 into the arc complex of the surface and use this to show that thi…

Counting subgroups of a surface using convex core lengths.

Given any generating set of any pseudo-Anosov-containing subgroup of the mapping class group of a surface, we construct a pseudo-Anosov with word length bounded by a constant depending only on the surface. More generally, in any subgroup G we find an element f with the property that the minimal subsurface supporting a …

Johnson kernel generated by specific Dehn twists on surfaces.

Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.

The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.

Study classifies mapping class groups with hyperbolic actions on infinite-type surfaces.

Combination theorem for PGF groups helps in constructing new examples and understanding their geometry.

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, w…

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The extension problem in Torelli topology is the problem of determining when a diffeomorphism of compact connected subsurface of a closed surface can be extended to a diffeomorphism of t…

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this ca…

The contraction of the image of the Johnson homomorphism is called the Chillingworth class. In this paper, we derive a combinatorial description of the Chillingworth class for Putman's subsurface Torelli groups. We also prove the naturality and uniqueness properties of the map whose image is the dual of the Chillingwor…

Study on compact and finite-type support in mapping class group homology.

Efficiently quantifies uncertainty in subsurface flow using neural networks guided by theory.

Subsurface applications including geothermal, geological carbon sequestration, oil and gas, etc., typically involve maximizing either the extraction of energy or the storage of fluids. Characterizing the subsurface is extremely complex due to heterogeneity and anisotropy. Due to this complexity, there are uncertainties…

SAGE generates subsurface velocity models from sparse well logs and seismic images.

Let $S$ be an infinite-type surface and $p\in S$. We show that the Thurston-Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of $Mod(S;p)$ on the loop graph $L(S;p)$ that do not leave any finite-type subsurface $S'\subset S$ i…

We introduce machinery to allow ``cut-and-paste''-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives different groups depending on how the subsurfaces are embedded. We define a categor…

Consider the mapping class group $\Mod_{g,p}$ of a surface $Σ_{g,p}$ of genus $g$ with $p$ punctures, and a finite collection $\{f_1,...,f_k\}$ of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prov…

Study arcs on surfaces, focusing on topological aspects and group actions.

Combines deep generative models with ensemble methods for subsurface property estimation.

Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes αand βon the boundary of M. The method applies when βis the boundary slope of an essential surface F …

We compare the flat geometry associated to a quadratic differential with the hyperbolic geometry associated to the underlying Riemann surface. We show that if a curve is contained in a thick subsurface, then its hyperbolic length is comparable to its flat length times the flat size of the subsurface.

The study finds abundant normal generators for mapping class groups.

New projection complex shows some surface homeomorphisms have positive commutator length.

This paper has two main goals. First, we give a complete, explicit, and computable solution to the problem of when two simple closed curves on a surface are equivalent under the Johnson kernel. Second, we show that the Johnson filtration and the Johnson homomorphism can be defined intrinsically on subsurfaces and prove…

Study block mapping class groups and their finiteness properties.

We study the connections between subsurface projections in curve and arc complexes in fibered 3-manifolds and Agol's veering triangulation. The main theme is that large-distance subsurfaces in fibers are associated to large simplicial regions in the veering triangulation, and this correspondence holds uniformly for all…

Formula for projecting geodesics in hyperbolic 3-manifolds, relating lengths to subsurface projections.

Classification of groups as symmetries of infinite translation surfaces.

New graphs show hierarchical hyperbolic properties, extending previous work.

Graphs of multicurves are hyperbolic, relatively hyperbolic, or thick.

PIML enhances machine learning for subsurface energy systems.

SURGIN uses generative models to infer subsurface flow data efficiently.

We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This result is an important step in the proof that the disk complex is Gromov hyperbolic…

Proves conjecture about surface cover homology.

Uniform bound on geodesic images for surfaces using bicorn curves.

The article studies Hamiltonian flows on surface group representations induced by invariant multi-functions.

Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomia…

Tackles dynamic subsurface flow via GAN with physical theory constraints.

In this paper, by putting a separating incompressible surface in a 3-manifold into Morse position relative to the height function associated to a strongly irreducible Heegaard splitting, we show that an incompressible subsurface of the Heegaard splitting can be found, by decomposing the 3-manifold along the separating …

Develops active intervals for geodesics in Teichmüller space.

Growth rates of geodesics on modular orbifolds are studied.