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Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groupsAug 21 2019This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms ... More
Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifoldsAug 20 2019We study the Smith forms of matrices of the form $f(C_g)$ where $f(t),g(t)\in R[t]$, $C_g$ is the companion matrix of the (monic) polynomial $g(t)$, and $R$ is an elementary divisor domain. Prominent examples of such matrices are circulant matrices, skew-circulant ... More
Optimal Control for Chemotaxis Systems and Adjoint-Based Optimization with Multiple-Relaxation-Time Lattice Boltzmann ModelsAug 11 2019This paper is devoted to continuous and discrete adjoint-based optimization approaches for optimal control problems governed by an important class of Nonlinear Coupled Anisotropic Convection-Diffusion Chemotaxis-type System (NCACDCS). This study is motivated ... More
Two generalisations of Leighton's TheoremAug 02 2019Leighton's graph covering theorem says that two finite graphs with a common cover have a common finite cover. We present a new proof of this using groupoids, and use this as a model to prove two generalisations of the theorem. The first generalisation, ... More
Quillen's conjecture for groups of p-rank 3Jul 03 2019Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any group $G$ ... More
Minimal resolutions of monomial idealsJun 20 2019An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional generalizations of combined ... More
Homology, lower central series, and hyperplane arrangementsJun 12 2019We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. ... More
A topological model for the coloured Alexander invariantsJun 10 2019Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. This sequence recovers the original Alexander polynomial as the first term. ... More
Arithmetic topology in Ihara theory II: Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbolsJun 03 2019We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. ... More
$F$-polynomials of tabulated virtual knotsJun 02 2019A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By the construction, ... More
Volumes of random 3-manifoldsMay 13 2019We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.
Volumes of random 3-manifoldsMay 13 2019Jul 29 2019We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.
The geometry of groups containing almost normal subgroupsMay 08 2019A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse ... More
Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$Apr 30 2019Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ ... More
Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$Apr 30 2019Jun 24 2019Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ ... More
Controlled surgery and $\mathbb{L}$-homologyApr 29 2019This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $\sigma^c (f, ... More
A proof of the refined PRV conjecture via the cyclic convolution varietyApr 25 2019In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar's refinement. The proof involves ... More
The winding invariantApr 22 2019Every element $w$ in the commutator subgroup of the free group $\mathbb{F}_2$ of rank 2 determines a closed curve in the grid $\mathbb{Z} \times \mathbb{R} \cup \mathbb{R} \times \mathbb{Z} \subseteq \mathbb{R}^2$. The winding numbers of this curve around ... More
Semi-equivelar maps of Euler characteristics -2 with few verticesApr 16 2019We enumerate and classify all the semi equivelar maps on the surface of $ \chi=-2 $ with up to 12 vertices. We also determine which of these are vertex-transitive and which are not.
Relative Combinatorial AsphericityApr 16 2019We introduce relative versions of diagrammatic reducibility (DR) and vertex asphericity (VA). The definition of diagrammatic reducibility of a 2-complex goes back to Sieradski 1983 who developed it as a tool for detecting asphericity of a 2-complex. We ... More
The spectrum of simplicial volumeApr 09 2019New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in ... More
Overcommuting pairs in groups and 3-manifolds bounding themMar 27 2019We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3-manifold and their Heegaard genus. We give many examples including translations ... More
Regularity of limit sets of Anosov representationsMar 26 2019Apr 09 2019In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the ... More
Regularity of limit sets of Anosov representationsMar 26 2019In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the ... More
Directed diagrammatic reducibilityMar 11 2019We introduce the notion of directed diagrammatic reducibility which is a relative version of diagrammatic reducibility. Directed diagrammatic reducibility has strong group theoretic and topological consequences. A multi-relator version of the Freiheitssatz ... More
On the top homology group of Johnson kernelMar 09 2019The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are ... More
Isotopy and equivalence of closed surface braidsMar 05 2019The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to equivalence and isotopy (with a small indeterminacy for isotopy of closed sphere braids), in terms of the algebra ... More
Isotopy and homeomorphism of closed surface braidsMar 05 2019Apr 29 2019The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), in terms of the algebra ... More
Essential loops in taut ideal triangulationsFeb 08 2019In this note we combinatorialise a technique of Novikov. We use this to prove that, in a three-manifold equipped with a taut ideal triangulation, any vertical or normal loop is essential in the fundamental group.
Obstructions for automorphic quasiregular maps and Lattès-type uniformly quasiregular mapsFeb 01 2019Suppose that $M$ is a closed, connected, and oriented Riemannian $n$-manifold, $f \colon \mathbb{R}^n \to M$ is a quasiregular map automorphic under a discrete group $\Gamma$ of Euclidean isometries, and $f$ has finite multiplicity in a fundamental cell ... More
An exotic presentation of Q_28Jan 30 2019We introduce a new family of presentations for the quaternion groups and show that for the quaternion group of order 28, one of these presentations has non-standard second homotopy group.
Jones representations of Thompson's group $F$ arising from Temperley-Lieb-Jones algebrasJan 29 2019Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group $F$ equipped with canonical (vacuum) vectors and ... More
On the finiteness length of some soluble linear groupsJan 20 2019Let $R$ be a commutative ring with unity. We prove that the finiteness length of a group $G$ is bounded above by the finiteness length of the Borel subgroup of rank one $\mathbf{B}_2^\circ(R) = \left( \begin{smallmatrix} * & * \\ 0 & * \end{smallmatrix} ... More
Realizable ranks of joins and intersections of subgroups in free groupsJan 14 2019The famous Hanna Neumann Theorem gives an upper bound for the ranks of the intersection of arbitrary subgroups $H$ and $K$ of a non-abelian free group. It is an interesting question to "quantify" this bound with respect to the rank of $H\vee K$, the subgroup ... More
Realizable ranks of joins and intersections of subgroups in free groupsJan 14 2019Feb 11 2019The famous Hanna Neumann Conjecture (now the Friedman-Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups $H$ and $K$ of a non-abelian free group. It is an interesting question to `quantify' this bound with respect ... More
Higher generating subgroups and Cohen-Macaulay complexesJan 14 2019Jan 31 2019We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe ... More
Higher generating subgroups and Cohen-Macaulay complexesJan 14 2019Aug 02 2019We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe ... More
Intrinsic linking and knotting in tournamentsJan 11 2019A directed graph $G$ is $\textit{intrinsically linked}$ if every embedding of that graph contains a non-split link $L$, where each component of $L$ is a consistently oriented cycle in $G$. A $\textit{tournament}$ is a directed graph where each pair of ... More
The Torsion Generating Set Of The Extended Mapping Class Groups In Low Genus CasesJan 05 2019We prove that for genus $g=3,4$, the extended mapping class group $\text{Mod}^{\pm}(S_g)$ can be generated by two elements of finite orders. But for $g=1$, $\text{Mod}^{\pm}(S_1)$ cannot be generated by two elements of finite orders.
Intrinsic linking and knotting are arbitrarily complex in directed graphsJan 04 2019Fleming and Foisy recently proved the existence of a digraph whose every embedding contains a $4$-component link, and left open the possibility that a directed graph with an intrinsic $n$-component link might exist. We show that, indeed, this is the case. ... More
Cubulating one-relator products with torsionDec 22 2018Jan 11 2019We generalize results of Lauer and Wise to show that a one-relator product of locally indicable groups whose defining relator has exponent at least 4 admits a proper and cocompact action on a CAT(0) cube complex if the factors do.
Bowditch's Q-conditions and Minsky's primitive stabilityDec 11 2018For the action of the outer automorphism group of the rank two free group on the corresponding variety of PSL(2,C) characters, two domains of discontinuity have been known to exist that are strictly larger than the set of Schottky characters. One is introduced ... More
Affine actions with Hitchin linear partDec 10 2018Properly discontinuous actions of a surface group by affine automorphisms of $\mathbb R^d$ were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then ... More
$2$-stratifolds with fundamental group $\mathbb{Z}$Dec 04 2018$2$-stratifolds are a generalization of $2$-manifolds that occur as objects in applications such as in TDA. These spaces can be described by an associated bicoloured labelled graph. In previous papers we obtained a classification of 1-connected trivalent ... More
Subgroups of word hyperbolic groups in rational dimension 2Nov 22 2018Jan 15 2019A result of Gersten states that if $G$ is a hyperbolic group with integral cohomological dimension $\mathsf{cd}_{\mathbb{Z}}(G)=2$ then every finitely presented subgroup is hyperbolic. We generalize this result for the rational case $\mathsf{cd}_{\mathbb{Q}}(G)=2$. ... More
Generalized torsion and decomposition of 3-manifoldsNov 19 2018A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion ... More
On self-affine tiles whose boundary is a sphereNov 16 2018Jun 20 2019Let $M$ be a $3\times 3$ integer matrix each of whose eigenvalues is greater than $1$ in modulus and let $\mathcal{D}\subset\mathbb{Z}^3$ be a set with $|\mathcal{D}|=|\det M|$, called digit set. The set equation $MT = T+\mathcal{D}$ uniquely defines ... More
The Torsion Generating Set Of The Dehn Twist Subgroups Of Non-orientable SurfacesNov 12 2018Nov 19 2018Let $N_g$ be the non-orientable surface with genus $g$, $\text{MCG}(N_g)$ be the mapping class group of $N_g$, $\mathcal{T}(N_g)$ be the index 2 subgroup generated by all Dehn twists of $\text{MCG}(N_g)$. We prove that for odd genus, $\mathcal{T}(N_g)$ ... More
Balanced presentations for fundamental groups of curves over finite fieldsNov 10 2018We show that the algebraic fundamental group of a smooth projective curve over a finite field admits a finite topological presentation where the number of relations does not exceed the number of generators.
Homotopy type of the complex of free factors of a free groupOct 22 2018May 22 2019We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent ... More
Homotopy type of the complex of free factors of a free groupOct 22 2018We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent ... More
Rigidity of the saddle connection complexOct 01 2018Nov 19 2018For a half-translation surface (S,q), the associated saddle connection complex A(S,q) is the simplicial complex where vertices are the saddle connections on (S,q), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can ... More
Many Three Dimensional Objects Inspired From Finite GroupsSep 24 2018Starting from considering deeper relationship between conjugacy classes and irreducible representations of a finite group $G$, we find some quite simple $R-$matrice defined by using finite groups. This construction produces many sets (or topological spaces) ... More
Aspects of the topology and combinatorics of Higgs bundle moduli spacesSep 15 2018Dec 08 2018This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space ... More
Aspherical Relative Presentations All Over AgainSep 10 2018Nov 21 2018The concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. ... More
The continuity of Darboux injections between manifoldsSep 02 2018Sep 12 2018We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ ... More
Quantum computing with Bianchi groupsAug 21 2018Dec 13 2018It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing $d$-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups ... More
Ineffectiveness of homotopical invariants on Nakanishi's 4-move conjectureAug 16 2018Aug 20 2018A $4$-move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove that any invariant ... More
A simpler $\mathbb R^3$ realization of the Möbius stripAug 12 2018A very simple $\mathbb R^3$ realization of the M\"obius strip, significantly simpler than the common one, is given. For any, however large width/length ratio of the strip, it is shown that this realization, in contrast with the common one, is the union ... More
Higher Order Degrees of Affine Plane Curve ComplementsAug 09 2018We study finiteness (and vanishing) properties of the higher order degrees associated to complements of complex affine plane curves with mild singularities at infinity. Our results impose new obstructions on the class of groups that can be realized as ... More
Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal SurfacesAug 02 2018We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial ... More
Isotopic tiling theory for hyperbolic surfacesAug 02 2018Dec 20 2018In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial ... More
On type-preserving representations of the thrice punctured projective plane groupJul 22 2018In this paper we consider type-preserving representations of the fundamental group of the three--holed projective plane into $\mathrm{PGL}(2, \R) =\mathrm{Isom}(\HH^2)$ and study the connected components with non-maximal euler class. We show that in euler ... More
Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental groupJul 10 2018Feb 08 2019Let $S=\Gamma\backslash \mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $\Gamma \le \operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider any generating set $\mathfrak S$ of $\Gamma$. When sampling ... More
Realisation of groups as automorphism groups in categoriesJul 02 2018Oct 15 2018It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely ... More
A counterexample to a strong version of the Andrews-Curtis conjectureJun 29 2018We prove that the presentations $\langle x,y | [x,y],1 \rangle$ and $\langle x,y | [x,[x,y^{-1}]]^2y[y^{-1},x]y^{-1},[x,[[y^{-1},x],x]] \rangle$ are not $Q^*$-equivalent even though their standard complexes have the same simple homotopy type.
Complete regular dessins and skew-morphisms of cyclic groupsJun 19 2018A dessin is a 2-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular ... More
Maximal subgroups of the modular and other groupsJun 11 2018In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${\rm PSL}_2(\mathbb Z)\cong C_3*C_2$ are ... More
Crossing Matrices of Positive BraidsMay 30 2018The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure ... More
On different notions of calibrations for minimal partitions and minimal networks in $\mathbb{R}^2$May 29 2018Apr 11 2019Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of Steiner problem in several variants. Our goal is to compare the different notions ... More
Automorphism groups of maps, hypermaps and dessinsMay 24 2018A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism ... More
Measure equivalence for non-unimodular groupsMay 23 2018We undertake a comprehensive study of measure equivalence between general locally compact, second countable groups, providing operator algebraic and ergodic theoretic reformulations, and complete the classification of amenable groups within this class ... More
A characterization on separable subgroups of 3-manifold groupsMay 22 2018In this paper, we give a complete characterization on which finitely generated subgroups of finitely generated $3$-manifold groups are separable. Our characterization generalizes Liu's spirality character on $\pi_1$-injective immersed surface subgroups ... More
Knot-theoretic ternary groupsMay 20 2018Aug 05 2018We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct ... More
Models of Simply-connected Trivalent $2$-dimensional Stratifolds with an Implementation CodeMay 14 2018Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We develop operations on their associated labeled graphs that will effectively construct from a single vertex all ... More
Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensionsMay 09 2018We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is ... More
Areas of spherical polyhedral surfaces with regular facesApr 30 2018For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of planar graphs ... More
Mapping class groups of covers with boundary and braid group embeddingsApr 27 2018We consider finite-sheeted, regular, possibly branched covering spaces of surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping ... More
Semi-free subgroups of a profinite surface groupApr 20 2018We show that every closed normal subgroup of infinite index in a profinite surface group $\Gamma$ is contained in a semi-free profinite normal subgroup of $\Gamma$. This answers a question of Bary-Soroker, Stevenson, and Zalesskii.
Topics in Geometric Group Theory. Part IApr 14 2018This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of easy groups. ... More
Signatures of surface bundles and stable commutator lengths of Dehn twistsApr 05 2018The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which ... More
Space of isospectral periodic tridiagonal matricesMar 30 2018Dec 14 2018A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized Shr\"{o}dinger ... More
All hyperbolic Coxeter $n$-cubesMar 28 2018Beside simplices, $n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter $n$-cubes are not classified. We show that there is no hyperbolic Coxeter $n$-cube for $n\geq~6$, and provide a full classification ... More
On infinitely generated homology of Torelli groupsMar 25 2018Let $\mathcal{I}_g$ be the Torelli group of an oriented closed surface $S_g$ of genus $g$, that is, the kernel of the action of the mapping class group on the first integral homology group of $S_g$. We prove that the $k$th integral homology group of $\mathcal{I}_g$ ... More
On infinitely generated homology of Torelli groupsMar 25 2018Mar 26 2019Let $\mathcal{I}_g$ be the Torelli group of an oriented closed surface $S_g$ of genus $g$, that is, the kernel of the action of the mapping class group on the first integral homology group of $S_g$. We prove that the $k$th integral homology group of $\mathcal{I}_g$ ... More
The relationship of generalized manifolds to Poincaré duality complexes and topological manifoldsMar 23 2018The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincar\'{e} duality complexes (PD complexes). The problem is that an arbitrary generalized manifold $X$ is always an ENR space, but it is not necessarily ... More
Commensurating actions for groups of piecewise continuous transformationsMar 22 2018We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the ... More
Profinite completions, cohomology and JSJ decompositions of compact 3-manifoldsFeb 26 2018In this paper we extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary. We also illustrate an alternative and perhaps more ... More
Toric topology of the complex Grassmann manifoldsFeb 18 2018Feb 28 2018The family of complex Grassmann manifolds $G_{n,k}$ with the canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogous of the moment map $\mu : G_{n,k}\to \Delta_{n,k}$ for the hypersimplex $\Delta_{n,k}$, is well known. In this paper we study ... More
Toric topology of the complex Grassmann manifoldsFeb 18 2018Apr 02 2019The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In this paper we ... More
Toric topology of the complex Grassmann manifoldsFeb 18 2018Jul 14 2019The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In this paper we ... More
Unitary Group Integrals, Surfaces, and Mapping Class GroupsFeb 13 2018Since the 1970's, Physicists and Mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: ... More
Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$Feb 13 2018May 03 2019Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: ... More
Universal quantum computing and three-manifoldsFeb 12 2018Nov 23 2018A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of $3$-manifolds. A magic state and the Pauli ... More
A new construction of CAT(0) cube complexesFeb 06 2018We introduce the notion of cube complex with coupled link (CLCC) as a mean of constructing interesting CAT(0) cubulated groups. CLCCs are defined locally, making them a useful tool to use when precise control over the links is required. In this paper ... More
Some remarks on PL collapsible covers of 2-dimensional polyhedraFeb 05 2018We analyze the topology and geometry of a polyhedron of dimension 2 according to the minimum size of a cover by PL collapsible polyhedra. We provide partial characterizations of the polyhedra of dimension 2 that can be decomposed as the union of two PL ... More
Topological aspect of monodromy groupoid for a group-groupoidJan 26 2018In this paper we develop star topological and topological group-groupoid structures of monodromy groupoid and prove that the monodromy groupoid of a topological group-groupoid is also a topological group-groupoid.
Schubert Decomposition for Milnor Fibers of the Varieties of Singular MatricesJan 21 2018We consider the varieties of singular $m \times m$ complex matrices which may be either general, symmetric or skew-symmetric (with $m$ even). For these varieties we have shown in another paper that they had compact "model submanifolds", for the homotopy ... More
Schubert Decomposition for Milnor Fibers of the Varieties of Singular MatricesJan 21 2018Sep 18 2018We consider the varieties of singular $m \times m$ complex matrices which may be either general, symmetric or skew-symmetric (with $m$ even). For these varieties we have shown in another paper that they had compact "model submanifolds", for the homotopy ... More
Profinite rigidity of graph manifolds, II: knots and mapping classesJan 19 2018Feb 09 2018In 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 ... More