Every infinite order mapping class has an infinite order action on the homology of some finite cover.
problem Proving every infinite order mapping class has an infinite order action on the homology of some finite cover.
method Theory of homological shadows and Fourier analysis.
result There exists a finite solvable cover and a lift of an infinite order mapping class with an infinite order action on the homology.
A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of l…
We consider the space of all representations of the commutator subgroup of a knot group into a finite abelian group Σ, together with a shift map σ_x. This is a finite dynamical system, introduced by D.Silver and S. Williams. We describe the lengths of its cycles in terms of the roots of the Alexander polynomial of the …
Counterexample found for Stein property of certain solvable Lie groups.
problem Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.
method Constructing a solvable Lie group with specific properties.
result A simply connected solvable Lie group with a left-invariant complex structure whose universal cover is not Stein.
Study of affine transformations on topological manifolds, focusing on local freeness and solvability.
problem Understanding the action of affine transformations on topological manifolds.
method Analyzing the subgroup of homeomorphisms that lift to affine transformations and studying the resulting foliation.
result The connected component of the subgroup acts locally freely and is solvable, with additional properties for polynomial manifolds.
New method finds unbranched covers with non-kernel homology.
problem Finding unbranched covers with non-kernel homology.
method Explains existing approach of Malestein and Putman, simplifies and generalizes it.
result Existence of unbranched covers with non-kernel homology.
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic…
Dimer coverings (or perfect matchings) of a finite graph are classical objects of graph theory appearing in the study of exactly solvable models of statistical mechanics. We introduce more general dimer labelings which form a topological space called the dimer space of the graph. This space turns out to be a cubed comp…
New representation shows non virtually solvable subgroups of mapping class groups have non virtually solvable elements.
problem Understanding non virtually solvable subgroups of mapping class groups.
method Finite dimensional homological representation of Mod(Σ) to show non virtually solvable elements.
result There exists a finite dimensional homological representation of Mod(Σ) such that the image of a non virtually solvable subgroup is also non virtually solvable.
A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π,1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple suff…
We present a method for computing the number of epimorphisms from a finitely-presented group G to a finite solvable group Γ, which generalizes a formula of Gäschutz. Key to this approach are the degree 1 and 2 cohomology groups of G, with certain twisted coefficients. As an application, we count low-index subgroups of …
New proof shows RFRS groups virtually fibred if their L2-Betti number vanishes.
problem Characterizing residually finite rationally solvable groups.
method Using L2-Betti numbers to prove virtual fibering. result Finitely generated RFRS groups are virtually fibred if their L2-Betti number vanishes. The paper classifies almost complex manifolds with non-degenerate torsion and their coverings.
problem Classification of almost complex manifolds with non-degenerate torsion.
method Analysis of homogeneous and non-homogeneous manifolds, classification of Lie algebras, covering spaces.
result Classification of manifolds with solvable Lie algebra up to coverings.
Suppose G is a connected complex Lie group and Γ is a discrete subgroup such that X:=G/Γ is Kähler and the codimension of the top non--vanishing homology group of X with coefficients in Z2 is less than or equal to two. We show that G is solvable and a finite covering of X is biholomorphic to a …
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finite…
We show that the Membership Problem for finitely generated subgroups of 3-manifold groups is solvable.
New examples of Laplacian solitons found on solvable Lie groups.
problem Finding closed Laplacian solitons with finite-time singularities.
method Left-invariant G2-structures on solvable Lie groups.
result First examples of closed Laplacian solitons with finite-time singularities.
It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
The fundamental n-quandles of links are residually finite for n ≥ 2.
problem Residual finiteness of fundamental n-quandles of oriented links.
method Investigation of residual finiteness and subquandle separability of quandles; use of Winker's work on 3-sphere branched covers.
result Fundamental n-quandles of oriented links are residually finite for each n ≥ 2.
We define new higher-order Alexander modules An(C) and higher-order degrees δn(C) which are invariants of the algebraic planar curve C. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve C. These invariants are in the spirit of th…
A companion result of the the Tits alternative for Out(Fn) is proved: Every solvable subgroup of Out(Fn) is finitely generated and virtually abelian.
Local description of solvable Lie algebras of vector fields.
problem Understanding solvable Lie algebras of vector fields.
method Local and constructive differential geometric description.
result Implication of Lie's conjecture for solvable Lie algebras.
Proves conditions for nearby special Lagrangians in Calabi-Yau manifolds.
problem Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
method Analyzes fundamental group, Floer cohomology, and special Lagrangian conditions.
result Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of $\Aut(G)$ on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev…
In the paper we consider the following conjecture: if a finite group G possesses a solvable π-Hall subgroup H, then there exist elements x,y,z,t∈G such that the identity H∩Hx∩Hy∩Hz∩Ht=Oπ(G) holds. The minimal counter example is shown to be an almost simple group of Lie type.
Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this asser…
Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.
problem Characterizing solvable symplectic Lie algebras.
method Symplectic double extension process.
result Classifies Lie algebras of dimensions up to 6 and proves structural theorems.
Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.
problem Investigate the existence of complex curves in hypercomplex nilmanifolds.
method Analyze quaternionic-solvable hypercomplex structures on nilpotent Lie algebras and prove the absence of complex curves in complex manifolds.
result Prove the non-existence of complex curves in complex manifolds associated with quaternionic-solvable hypercomplex structures.
We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such a…
The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.
problem Characterizing when numerical criteria for PDE solvability fail.
method Finite number of subvarieties violating Nakai type criterion, and their rigidity.
result Finite number of subvarieties violating the Nakai type criterion, and these subvarieties are rigid.
3-manifold groups' word problem solved in nearly linear time.
problem Solving the word problem for 3-manifold groups efficiently.
method Proof for admissible graphs of groups, leveraging Croke and Kleiner's work.
result Word problem solved in O(nlogn) time for 3-manifold groups. The study examines Lorentzian metrics on specific Lie groups and their curvatures.
problem Analyzing Lorentzian metrics on specific Lie groups and their geometric properties.
method Explicitly lists all Lorentzian left invariant metrics and computes their curvatures.
result Characterization of metrics with constant curvature, semi-symmetric non locally symmetric, and Ricci solitons.
Complete list of solvable BS groups' actions on hyperbolic spaces.
problem Characterizing actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces.
method Complete enumeration and classification of actions up to equivalence.
result Finitely many equivalence classes of actions, each containing a point, tree, or hyperbolic plane action.
Compact Einstein manifolds of low dimensions are studied based on Lie group actions.
problem Characterize compact pseudo-Riemannian homogeneous Einstein manifolds of low dimensions.
method Analyze Lie group actions and bilinear forms on Lie algebras.
result In dimensions ≤ 7, compact quotients exist only for nilpotent groups.
Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M, and…
Study proves surface group quotients have non-virtually solvable profinite completions.
problem Understanding profinite completions of surface group quotients.
method Quantum representations of mapping class groups.
result Profinite completions are not virtually solvable, with infinitely many finite simple quotients.
Study geometric actions of groups on horocyclic products.
problem Understanding geometric actions of groups on horocyclic products.
method Analyzing geometric actions of groups on horocyclic products of CAT(-κ) spaces.
result Groups acting on horocyclic products are either ascending HNN extensions of finitely-generated virtually nilpotent groups or not finitely presented.
This paper extends Alexander invariants to higher-dimensional hypersurface complements.
problem Understanding the topology of higher-dimensional hypersurface complements.
method Defining higher-order Alexander modules and degrees from solvable covers of hypersurface complements.
result Higher-order degrees of hypersurface complements are bounded by a linear combination of local link pairs.
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to R⋉Rn wh…
New Lie algebras from quivers lead to rigid Ricci solitons.
problem Constructing Lie algebras from quivers to study geometric structures.
method Using finite quivers without cycles to construct solvable Lie algebras and proving their geometric properties.
result Simply-connected Lie groups corresponding to these Lie algebras admit left-invariant Ricci solitons, and when quivers are oriented multi-trees, these groups are rigid.
Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
problem Characterizing simply connected indefinite homogeneous spaces of finite volume.
method Analyzing Lie algebras with abelian solvable radical and symmetric bilinear form.
result Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant variables) on specific finite-dimensional subspaces of the space of generalized symmetries of some system of partial differential equations, we explicitly determine the dependance of coefficients of generalized symmetries fro…
A closed 4-manifold (or, more generally, a finite PD4-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a PD3-complex.
This work is focused on the solvability of initial-boundary value problems for degenerate parabolic partial differential equations that arise in the pricing of Asian options, and on the investigation of differential and certain qualitative properties of solutions of such equations. The generalized solvability for such …
Study covers of Chamanara surface with large Veech groups.
problem Characterize finite abelian covers with large Veech groups.
method Analyze Veech groups of finite abelian covers of the Chamanara surface.
result Every free group arises as the projective Veech group of a finite-area infinite translation surface.
The paper characterizes geometrically finite surfaces via geodesic covers.
problem Characterizing geometrically finite surfaces.
method Study of geodesic covers of Fuchsian groups and their metric properties.
result Finiteness of geodesic covers characterizes geometrically finiteness.