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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.

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

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

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

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 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

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

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

Moduli spaces of real projective structures on surfaces: Notes on a paper by V.V. Fock and A.B. GoncharovJan 11 2018These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharov's description of ... More

The Action of Young Subgroups on the Partition ComplexJan 04 2018Mar 06 2018We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_n|$, which is the $\Sigma_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed ... More

Remarks on multisymplectic reductionDec 28 2017Jan 15 2018The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries.

Measuring complexity of curves on surfacesDec 18 2017Jul 18 2018We consider the relations between different measures of complexity for free homotopy classes of curves on a surface $\Sigma$, including the minimum number of self-intersections, the minimum length of the words representing them in a geometric presentation ... More

A combinatorial description of the centralizer algebras connected to the Links-Gould InvariantDec 13 2017In this paper we study the tensor powers of a $4$-dimensional representation of the quantum super-algebra $U_q(sl(2|1)$, focusing on the rings of its algebra endomorphisms so called centralizer algebras, denoted by $LG_n$. Their dimensions were conjectured ... More

A combinatorial description of the centralizer algebras connected to the Links-Gould InvariantDec 13 2017Mar 02 2019In this paper we study the tensor powers of the standard representation of the quantum super-algebra $U_q(sl(2|1)$, focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by $LG_n$. Their dimensions were conjectured ... More

A Homological model for the coloured Jones polynomialsDec 13 2017In this paper we will present a homological model for Coloured Jones Polynomials. For each color $N \in \N$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homological classes in a covering of the configuration space ... More

The complex of partial bases of a free groupNov 27 2017We prove that the simplicial complex whose simplices are the nonempty partial bases of $\mathbb{F}_n$ is homotopy equivalent to a wedge of $(n-1)$-spheres. Moreover, we show that it is Cohen-Macaulay.

Four-dimensional graph-manifolds with fundamental groups quasi-isometric to fundamental groups of orthogonal graph-manifoldsNov 23 2017We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a universal cover ... More

Quadric ComplexesNov 15 2017Apr 09 2018Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2-dimensional CAT(0) cube complexes and are a square analog of systolic complexes. We introduce and study the basic properties ... More

Lifting classes for the fixed point theory of $n$-valued mapsNov 07 2017Sep 17 2018The theory of lifting classes of single-valued maps is extended to $n$-valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicot\'encatl. An equivalence ... More

Canonical measures on metric graphs and a Kazhdan's theoremNov 07 2017Nov 22 2018We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting ... More

Loxodromic elements in the cyclic splitting complex and their centralizersOct 28 2017We show that an outer automorphism acts loxodromically on the cyclic splitting complex if and only if it has a filling lamination and no generic leaf of the lamination is carried by a vertex group of a cyclic splitting. This is the analog for the cyclic ... More

Generating mapping class groups with elements of fixed finite orderOct 12 2017We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We additionally prove ... More

Relative cohomology theory for profinite groupsOct 02 2017In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite ... More

Intersection forms of almost-flat 4-manifoldsSep 14 2017We calculate intersection forms of all 4-dimensional almost-flat manifolds

Dehn functions of subgroups of right-angled Artin groupsSep 12 2017We show that for each positive integer $k$ there exist right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as $n^k$. As a consequence we produce examples of right-angled Artin groups containing finitely presented ... More

Brieskorn manifolds, generalised Sieradski groups, and coverings of lens spacesSep 11 2017The Brieskorn manifolds $B(p,q,r)$ are the $r$-fold cyclic coverings of the 3-sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic pre\-sen\-tation $G_{m}(w)$, where defining word ... More

Root extraction in one-relator groups and slendernessSep 09 2017In this note we strengthen a result of B. B. Newman and use it to prove a conjecture of J. Nakamura stated in \cite{Na} that torsion-free one-relator groups are noncommutatively slender.

Amalgamation and Symmetry: From Local to Global Consistency in The FiniteAug 31 2017Amalgamation patterns are specified by a finite collection of finite template structures together with a collection of partial isomorphisms between pairs of these. The template structures specify the local isomorphism types that occur in the desired amalgams; ... More

Isomorphisms between big mapping class groupsAug 28 2017Apr 30 2018We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping ... More

Subquandles of affine quandlesAug 08 2017Jun 05 2018A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming from the commutator ... More

Subquandles of affine quandlesAug 08 2017A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming from the commutator ... More

The infinite Fibonacci groups and relative asphericityAug 03 2017We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n) in {(7 + 5k,5), (8 + 5k,5)} where k is greater than or equal to 0. This together with previously known results yields a complete classification of the finite F(r,n), a problem ... More

2-stratifold spines of closed 3-manifoldsJul 18 2017$2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed branch curves. We obtain a list of all closed $3$-manifolds that have a $2$-stratifold as a spine.