arXiv research

The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.

The paper finds many thin subgroups isomorphic to Gromov-Piatetski-Shapiro lattices.

Study thin hyperbolic reflection groups and their properties.

Thin groups found in specific lattices.

In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it …

We consider a certain hybridization construction which produces a subgroup of ${\rm PU}(n,1)$ from a pair of lattices in ${\rm PU}(n-1,1)$. Among the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$, we show that the hybrid of pairs of Fuchsian subgroups ${\rm PU}(1,1,\mathcal{O}_d)$ is a lattice when $d=1$ and $d=7…

Abby Thompson proved that if a link $K$ is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin…

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible surfaces which are not tori. Next, such incompressible surfaces are related to …

Signature uniquely identifies piecewise linear surfaces up to thin homotopy.

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete. In this paper, let $K<Γ<G$ be an infinite normal subgroup of an arithmetic lattice $Γ$ in a rank one simple Lie group $G$, such that the quotient $Q=Γ/K$ is infinite. W…

If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…

New framework reveals limits of flexible, periodic thin surfaces.

New groups discovered with unique properties in a specific space.

Paper derives formulas for surface variations in shell theory.

Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and…

New proof of Haken's Lemma and Scharlemann's theorem using thin position.

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called thin position. Thin position is defined usin…

Let $T$ be a graph in a compact, orientable 3--manifold $M$ and let $Γ$ be a subgraph. $T$ can be placed in bridge position with respect to a Heegaard surface $H$. We show that if $H$ is what we call $(T,Γ)$-c-weakly reducible in the complement of $T$ then either a "degenerate" situation occurs or $H$ can be untelescop…

Constructs hyperbolic reflection groups with 3D limit sets.

We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful p…

We show that every thin position for a connected sum of small knots is obtained in an obvious way: place each summand in thin position so that no two summands intersect the same level surface, then connect the lowest minimum of each summand to the highest maximum of the adjacent summand below.

We show that if a knot or link has n thin levels when put in thin position then its exterior contains a collection of n disjoint, non-parallel, planar, meridional, essential surfaces. A corollary is that there are at least n/3 tetrahedra in any triangulation of the complement of such a knot.

Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $Φ(H)\neq H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $Φ(H)$ in $\mathrm{PSL}_2(\mathbb{R})$ is discrete. W…

This paper generalizes the definition of a Heegaard splitting to unify Scharlemann and Thomspon's concept of thin position for 3-manifolds, Gabai's thin position for knots, and Rubinstein's almost normal surface theory. This gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and Stocking. In the fi…

Wu has shown that if a link or a knot $L$ in $S^3$ in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that $L \subset S^3$ is prime, then the thin sphere of lowest width also does not have any vertical c…

The paper examines rigidity of thin domains under specific boundary conditions.

We extend the notion of thin multiple Heegaard splittings of a link in a 3-manifold to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly compressible bridge surface for a link K contained in a closed orientable irreducible 3-manifold M th…

Symmetries in shell theory lead to multiple deformation possibilities.

Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thi…

Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to $\{1,\dots, 1\}$, and give a relation between th…

Generalizes data thinning for various distributions.

The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.

Study explores kinematics of surfaces under metric restrictions.

Geometric proof of curve characterization using loop-bundles.

In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution and have the smallest possible rank. This allows for constructing a 3-periodic …

A new functional for simplicial surfaces is suggested. It is invariant with respect to Moebius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. as an application a bending energy for discrete thin-shells is derived.

Geometric bounds for low Steklov eigenvalues on hyperbolic surfaces with boundaries.

New lattices in higher dimensions have dense surface subgroups.

A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular l…

In this paper we determine a complete list of rational surface singularities which have metrically conical bilipschitz type of its inner metric. We achieve this by using the thick-thin decomposition of Birbrair, Neumann and Pichon.

The paper examines subgroup separability for surface and virtual braid groups.

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.

The study finds surface subgroups in cocompact lattices of $H^{2n}$ for $n\geq2$.

Minimal involutions generate a subgroup of nonorientable surfaces.

The paper controls the geometry of surface subgroups in specific Kleinian groups.

New combinatorial structures represent subgroups of surface groups, analogous to Stallings core graphs.