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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

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

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

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

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

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

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

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

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

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

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

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 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.

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

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

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 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

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.

Topological finiteness properties of monoids. Part 1: FoundationsJun 14 2017We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a ... More

On the peripheral subgroups of irreducible 3-manifold groups and acylindrical splittingsMay 17 2017Oct 19 2017We survey the problem of separation under conjugacy and malnormality of the abelian peripheral subgroups of an orientable, irreducible $3$-manifold $X$. We shall focus on the relation between this problem and the existence of acylindrical splittings of ... More

Generalized Fibonacci groups H(r,n,s) that are connected Labelled Oriented Graph groupsMay 16 2017Nov 07 2017The class of connected LOG (Labelled Oriented Graph) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in S^4, and so contains all knot groups. We investigate when Campbell and Robertson's ... More

Generating functions for the universal Hall-Littlewood $P$- and $Q$-functionsMay 13 2017Jul 22 2018Recently, P. Pragacz described the ordinary Hall-Littlewood $P$-polynomials by means of push-forwards (Gysin maps) from flag bundles in the ordinary cohomology theory. Together with L. Darondeau, he also gave push-forward formulas (Gysin formulas) for ... More

Normal closures of slope elements in knot groups and the peripheral Magnus propertyMay 12 2017Let K be a nontrivial knot in the 3-sphere with the exterior E(K). A slope element u in the knot group G(K) is a nontrivial element represented by a simple closed curve on the boundary of E(K). Each slope element u defines a normal subgroup, the normal ... More

Geometrically finite amalgamations of hyperbolic 3-manifold groups are not LERFMay 09 2017We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations ... More

A Pfaffian formula for the monomer-dimer model on surface graphsMay 02 2017We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises ... More

How to centralize and normalize quandle extensionsMar 27 2017Mar 29 2017We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization ... More

Measure equivalence and coarse equivalence for unimodular locally compact groupsMar 23 2017Sep 21 2018This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they admit free, ... More

Algebraic and topological properties of big mapping class groupsMar 08 2017Dec 01 2017Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, ... More

Graph complexity and Mahler measureJan 21 2017The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian group by graph ... More

Orientably-regular maps on twisted linear fractional groupsJan 20 2017We present an enumeration of orientably-regular maps with automorphism group isomorphic to the twisted linear fractional group $M(q^2)$ for any odd prime power $q$.

The Pachner graph of 2-spheresJan 18 2017Oct 10 2018It is well-known that the Pachner graph of $n$-vertex triangulated $2$-spheres is connected, i.e., each pair of $n$-vertex triangulated $2$-spheres can be turned into each other by a sequence of edge flips for each $n\geq 4$. In this article, we study ... More

A note on the relation between Hartnell's firefighter problem and growth of groupsJan 10 2017Jan 12 2017The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell. The game on a graph $G$ can be described as follows: let $f_n$ be a sequence of positive integers; an initial fire starts at a finite set of vertices; at ... More

Partition functions and a generalized coloring-flow duality for embedded graphsJan 02 2017Nov 07 2017Let $G$ be a finite group and $\chi: G \rightarrow \mathbb{C}$ a class function. Let $H = (V,E)$ be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection $F$ of faces of $H$. Define the partition ... More

Local and global coincidence homology classesDec 07 2016For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi- Bracci-Izawa-Suwa (2016) in the framework of Cech-de Rham cohomology. We take up ... More

Magnus pairs in, and free conjugacy separability of, limit groupsDec 01 2016There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable.

Species with potential arising from surfaces with orbifold points of order 2, Part II: arbitrary weightsNov 24 2016Apr 11 2017Let $\mathbf{\Sigma}=(\Sigma,M,O)$ be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and $\omega:O\rightarrow\{1,4\}$ a function. For each triangulation $\tau$ of $\mathbf{\Sigma}$ we construct ... More

Species with potential arising from surfaces with orbifold points of order 2, Part II: arbitrary weightsNov 24 2016Let (\Sigma,M,O) be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and let \omega be a function from O to {1,4}. For each triangulation \tau of (\Sigma,M,O) we construct a cochain complex ... More

Classification and Models of Simply-connected Trivalent $2$-dimensional StratifoldsNov 23 2016Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We obtain a classification of $1$-connected $2$-stratifolds in terms of their associated labeled graphs and develop ... More

Cyclically presented groups with length four positive relatorsNov 16 2016For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity for the underlying ... More

Cyclically presented groups with length four positive relatorsNov 16 2016Dec 20 2016For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity for the underlying ... More

Cyclically presented groups with length four positive relatorsNov 16 2016Nov 25 2016For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity for the underlying ... More

Finitely stable racks and rack representationsNov 14 2016We define a new class of racks, called finitely stable racks, which, to some extent, share various flavours with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct ... More

Finitely stable racks and rack representationsNov 14 2016Jun 23 2017We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct ... More

Hyperbolic jigsaws and families of pseudomodular groups INov 09 2016Jan 18 2017We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular ... More

Hyperbolic jigsaws and families of pseudomodular groups INov 09 2016We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular ... More

A small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twistsNov 02 2016We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn ... More

Conditions of smoothness of moduli spaces of flat connections and of representation varietiesOct 31 2016We use gauge theoretic and algebraic methods to reexamine the sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the representation variety of a finitely generated and presented group. In particular, ... More