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**Sosyoloji Konferansları (Istanbul Journal of Sociological Studies)**

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**Open Access journal**

ISSN (Print) 1304-0243

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**İstanbul Üniversitesi**[6 journals]

**An entropic characterization of the flat metrics on the two torus****Abstract:**Abstract The geodesic flow of the flat metric on a torus is minimizing the polynomial entropy among all geodesic flows on this torus. We prove here that this properties characterises the flat metric on the two torus.**PubDate:**2015-07-09

**On the volume conjecture for polyhedra****Abstract:**Abstract We formulate a generalization of the volume conjecture for planar graphs. Denoting by \(\langle \Gamma , c \rangle ^{\mathrm {U}}\) the Kauffman bracket of the graph \(\Gamma \) whose edges are decorated by real “colors” c, the conjecture states that, under suitable conditions, certain evaluations of \(\langle \Gamma ,\lfloor kc \rfloor \rangle ^{\mathrm {U}}\) grow exponentially as \(k\rightarrow \infty \) and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is \(\Gamma \) (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar graphs, generalizing the Gordon–Schulten recursion for the quantum 6j-symbols. Assuming that \(\langle \Gamma ,\lfloor kc \rfloor \rangle ^{\mathrm {U}}\) does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schläfli equation); moreover, any small perturbation of the volume function that is still a solution to these equations, is a perturbation by an additive constant. In the appendix we also provide a proof outlined elsewhere of the conjecture for an infinite family of planar graphs including the tetrahedra.**PubDate:**2015-07-08

**Distance two links****Abstract:**Abstract In this paper, we characterize all links in \(S^3\) with bridge number at least three that have a bridge sphere of distance two. We show that if a link L has a bridge sphere of distance at most two then it falls into at least one of three categories: The exterior of L contains an essential meridional sphere. L can be decomposed as a tangle product of a Montesinos tangle with an essential tangle in a way that respects the bridge surface and either the Montesinos tangle is rational or the essential tangle contains an incompressible, boundary-incompressible annulus. L is obtained by banding from another link \(L'\) that has a bridge sphere of the same Euler characteristic as the bridge sphere for L but of distance 0 or 1.**PubDate:**2015-07-07

**Congruence classes of points in quaternionic hyperbolic space****Abstract:**Abstract An important problem in quaternionic hyperbolic geometry is to classify ordered m-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, \(\overline{\mathbf{H}_{\mathbb H}^n}\) , up to congruence in the holomorphic isometry group \(\mathrm{PSp}(n,1)\) of \(\mathbf{H}_{\mathbb H}^n\) . In this paper we concentrate on two cases: \(m=3\) in \(\overline{\mathbf{H}_{\mathbb H}^n}\) and \(m=4\) on \(\partial \mathbf{H}_{\mathbb H}^n\) for \(n\ge 2\) . New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartan’s angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case.**PubDate:**2015-07-07

**Conjugacy limits of the diagonal cartan subgroup in**

$${\varvec{SL}}_\mathbf{3 }({\varvec{\mathbb {R}}})$$ S L 3 ( R )**Abstract:**Abstract A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan subgroup, \(C \le SL _3(\mathbb {R}).\) We prove a variant of a theorem of Haettel, and show that up to conjugacy, C has five possible conjugacy limit groups. Each conjugacy limit group is determined by a nonstandard triangle. We give a criterion for a sequence of conjugates of C to converge to each of the five conjugacy limit groups.**PubDate:**2015-07-02

**Constructing pseudo-Anosov maps with given dilatations****Abstract:**Abstract In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.**PubDate:**2015-07-01

**The JSJ-decompositions of one-relator groups with torsion****Abstract:**Abstract In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of one-relator groups with torsion closely resemble the structure of torsion-free hyperbolic groups.**PubDate:**2015-06-25

**Intersection theory of the Peterson variety and certain singularities of**

Schubert varieties**Abstract:**Abstract Precup recently proved that intersections with Schubert cells pave regular nilpotent Hessenberg varieties. We use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explicitly identify some of the coefficients of Schubert classes in this expansion, answering a problem of independent interest in Schubert calculus. We also identify some singular points in a certain family of Schubert varieties in general Lie type.**PubDate:**2015-06-21

**Subgroups of mapping class groups related to Heegaard splittings and**

bridge decompositions**Abstract:**Abstract Let \(M=H_1\cup _S H_2\) be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider the subgroup \({\text {MCG}}^0(H_j)\) of the mapping class group of \(H_j\) consisting of mapping classes represented by orientation-preserving auto-homeomorphisms of \(H_j\) homotopic to the identity, and let \(G_j\) be the subgroup of the automorphism group of the curve complex \(\mathcal {CC}(S)\) obtained as the image of \({\text {MCG}}^0(H_j)\) . Then the group \(G=\langle G_1, G_2\rangle \) generated by \(G_1\) and \(G_2\) acts on \(\mathcal {CC}(S)\) with each orbit being contained in a homotopy class in M. In this paper, we study the structure of the group G and examine whether a homotopy class can contain more than one orbit. We also show that the action of G on the projective lamination space of S has a non-empty domain of discontinuity when the Heegaard splitting satisfies R-bounded combinatorics and has high Hempel distance.**PubDate:**2015-06-18

**Compactifications of character varieties and skein relations on conformal**

blocks**Abstract:**Abstract Let \(M_C(G)\) be the moduli space of semistable principal G-bundles over a smooth curve C. We show that a flat degeneration of this space \(M_{C_{\Gamma }}(G)\) associated to a singular stable curve \(C_{\Gamma }\) contains the free group character variety \({\mathcal {X}}(F_g, G)\) as a dense open subset, where \(g = genus(C).\) In the case \(G = SL_2({\mathbb {C}})\) we describe the resulting compactification explicitly, and in turn we conclude that the coordinate ring of \(M_{C_{\Gamma }}(SL_2({\mathbb {C}}))\) is presented by homogeneous skein relations. Along the way, we prove the parabolic version of these results over stable, marked curves \((C_{\Gamma }, \vec {p}_{\Gamma })\) .**PubDate:**2015-06-18

**Line arrangements with the maximal number of triple points****Abstract:**Abstract The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields \(\mathbb F\) over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.**PubDate:**2015-06-16

**Coarse median structures and homomorphisms from Kazhdan groups****Abstract:**Abstract We study Bowditch’s notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan’s property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock–Druţu–Sapir on the mapping class groups of orientable surfaces. We discuss a metric approximation property of finite subsets in coarse median spaces extending the classical result on approximation of Gromov hyperbolic spaces by trees.**PubDate:**2015-06-13

**Quasi-arithmeticity of lattices in $${{\mathrm{PO}}}(n,1)$$ PO ( n , 1 )****Abstract:**Abstract We show that the non-arithmetic lattices in \({{\mathrm{PO}}}(n,1)\) of Belolipetsky and Thomson (Algebr Geom Topol 11(3):1455–1469, 2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and Piatetski-Shapiro are not quasi-arithmetic. A corollary of this is that there are, for all \(n\geqslant 2\) , non-arithmetic lattices in \({{\mathrm{PO}}}(n,1)\) that are not commensurable with the Gromov–Piatetski-Shapiro lattices.**PubDate:**2015-06-11

**Affine maps between CAT(0) spaces****Abstract:**Abstract We study affine maps between CAT(0) spaces with cocompact group actions, and show that they essentially split as products of dilations and linear maps (on the Euclidean factor). This extends known results from the Riemannian case. Furthermore, we prove a splitting lemma for the Tits boundary of a CAT(0) space with cocompact group action, a variant of a splitting lemma for geodesically complete CAT(1) spaces by Lytchak.**PubDate:**2015-06-07

**Metric behaviour of the Magnus embedding****Abstract:**Abstract The classic Magnus embedding is a very effective tool in the study of abelian extensions of a finitely generated group \(G\) , allowing us to see the extension as a subgroup of a wreath product of a free abelian group with \(G\) . In particular, the embedding has proved to be useful when studying free solvable groups. An equivalent geometric definition of the Magnus embedding is constructed and it is used to show that it is \(2\) -bi-Lipschitz, with respect to an obvious choice of generating sets. This is then applied to obtain a non-zero lower bound on \(L_p\) compression exponents in free solvable groups.**PubDate:**2015-06-01

**Invariance of finiteness of K-area under surgery****Abstract:**Abstract K-area is an invariant for Riemannian manifolds introduced by Gromov as an obstruction to the existence of positive scalar curvature. However in general it is difficult to determine whether K-area is finite or not in spite of its natural definition. In this paper, we study how the invariant changes under surgery.**PubDate:**2015-06-01

**On visualization of the linearity problem for mapping class groups of**

surfaces**Abstract:**Abstract We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describes the linearity for the mapping class group of compact orientable surface with boundary, up to center.**PubDate:**2015-06-01

**Every finite complex has the homology of some $${\mathrm{CAT}(0)}$$ CAT (**

0 ) cubical duality group**Abstract:**Abstract We prove that every finite connected simplicial complex has the homology of the classifying space for some \({\mathrm{CAT}(0)}\) cubical duality group. More specifically, for any finite simplicial complex \(X\) , we construct a locally \({\mathrm{CAT}(0)}\) cubical complex \(T_{X}\) and an acyclic map \(t_{X} : T_{X} \rightarrow X\) such that \(\pi _{1}(T_{X})\) is a duality group.**PubDate:**2015-06-01

**Euclidean hypersurfaces with a totally geodesic foliation of codimension**

one**Abstract:**Abstract We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered.**PubDate:**2015-06-01

**The Clifford index of line bundles on a 2-elementary K3 surface given by a**

double cover of a Del Pezzo surface**Abstract:**Abstract We call a K3 surface \(X\) a 2-elementary K3 surface if the Néron–Severi lattice \(S_X\) satisfies the condition that \(S_X^{*}/S_X\) is a 2-elementary group, where \(S_X^{*}:={{\mathrm{Hom}}}(S_X,\mathbb {Z})\) . In this paper, in the case where \(X\) is a 2-elementary K3 surface given by a double cover of a smooth Del Pezzo surface of degree \(4\le d\le 8\) and \(L\) is a base point free and big line bundle on \(X\) , for any smooth curve \(C\) in the linear system \( L \) , we investigate line bundles on \(C\) which compute the Clifford index of \(C\) .**PubDate:**2015-06-01