arXiv research

A universal branched 3-manifold $W$ characterizes Sol 3-manifolds.

Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.

3-manifolds with specific homology are cobordant if homeomorphic.

Study on achiral Sol 3-manifolds with density results.

The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We prove the conjecture in the case that the target 3-m…

3-manifold groups have a property that allows them to act on quasi-trees.

We classify all closed non-orientable P2-irreducible 3-manifolds with complexity up to 7, fixing two mistakes in our previous complexity-up-to-6 classification. We show that there is no such manifold with complexity less than 6, five with complexity 6 (the four flat ones and the filling of the Gieseking manifold, which…

We classify all closed non-orientable P2-irreducible 3-manifolds having complexity up to 6 and we describe some having complexity 7. We show in particular that there is no such manifold with complexity less than 6, and that those having complexity 6 are precisely the 4 flat non-orientable ones and the filling of the Gi…

Formula derived for volume entropy of certain metrics on Euclidean space.

We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CM…

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…

We compute the rings $H^*(N;\mathbb{F}_2)$ for $N$ a closed $\mathbb{S}ol^3$-manifold and then determine the Borsuk-Ulam indices $BU(N,φ)$ with $φ\not=0$ in $H^1(N;\mathbb{F}_2)$.

In this paper, we study strongly quasiconvex subgroups in a finitely generated $3$--manifold group $π_1(M)$. We prove that if $M$ is a compact, orientable $3$--manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $H \le π_1(M)$ has finite …

We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in $H^2 \times R$, $S^2 \times R$ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds ap…

The paper classifies harmonic and biharmonic submersions from Sol space.

Virtual reality explores non-Euclidean Sol geometry.

We show that the nonzero roots of the torsion polynomials associated to the infinite cyclic covers of a given compact, connected, orientable 3-manifold M are contained in a compact part of the complex plane a priori determined by M. This result is applied to prove that when M is closed, it dominates at most finitely ma…

We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in $SL(2,\Z)$ and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementa…

$Sol_3$ is shown not to be parabolic.

The paper studies geometric structures in Sol_3 with two connections.

Estimate sphere area in Sol group up to a factor of 10.

In this paper, we study biharmonic maps into Sol and Nil spaces, two model spaces of Thurston's 3-dimensional geometries. We characterize non-geodesic biharmonic curves in Sol space and prove that there exists no non-geodesic biharmonic helix in Sol space. We also show that a linear map from a Euclidean space into Sol …

In the homogeneous space Sol$_3$, a translation surface is parameterized by $x(s,t)=α(s)\astβ(t)$, where $α$ and $β$ are curves contained in coordinate planes and $\ast$ denotes the group operation of Sol$_3$. In this paper we study translation surfaces in Sol$_3$ whose mean curvature vanishes.

We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compact locally homogeneous Lorentz 3-manifolds with non-compact (local) isotropy group. One result is that, up to a finite cover, all such manifolds admit Lorentz metrics of (non-positive) constant sectionnal curvature. If the geometry is maximal, then the…

Formula found for surfaces in Sol_3, leading to gap results.

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$. The maximal compact subgroup of $Isom(Sol_1^4)$ is $D_4=\mathbb Z_4\rtimes\mathbb Z_2$. We shall exhibit an infra-solvmanifold with $Sol_1^4$-geometry whose holonomy is $D_4$. This implies that all …

The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.

SOL is an open-source library for scalable online learning algorithms, and is particularly suitable for learning with high-dimensional data. The library provides a family of regular and sparse online learning algorithms for large-scale binary and multi-class classification tasks with high efficiency, scalability, porta…

The paper classifies hypersurfaces in a specific 4D geometry.

Study on triharmonic curves in Sol space with constant curvature and torsion.

In this paper, we give complete classifications of linear $\infty$-harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all $\infty$-harmonic linear endomorphisms of Sol space and show that there is a subgroup of $\infty$-harmonic linear automorphisms in the group of linea…

Interpolates Sol geometry to Hyperbolic Space with a parameter.

In this paper, we study the Dirichlet problem for the minimal surface equation in $\rm Sol_3$ with possible infinite boundary data, where $\rm Sol_3$ is the non-abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometr…

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric…

Let $G$ be a connected Lie group and $Γ\subset G$ a lattice. Connection curves of the homogeneous space $M=G/Γ$ are the orbits of one parameter subgroups of $G$. To $block$ a pair of points $m_1,m_2 \in M$ is to find a finite set $B \subset M\setminus \{m_1, m_2 \}$ such that every connecting curve joining $m_1$ and $m…

Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.

In this paper we classify all surfaces in the 3-dimensional Lie group $Sol_3$ whose normals make constant angle with a left invariant vector field.

Survey on four-dimensional Thurston geometries with Riemannian metrics.

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive real…

Classifies Anosov flows on figure-eight knot surgeries.

The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.

Geodesics in Sol geometry described with invariant k and spiral properties.

In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].

The paper describes distances on Sol-type groups using novel geometric techniques.

We obtain an asymptotic formula for the spectrum distribution function of the Laplace operator on a compact Riemannian Sol-manifold in the adiabatic limit determined by a one-dimensional foliation defined by the orbits of a left-invariant flow.

The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov $φ:S\rightarrow S$ has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on…

The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift …