scholarly journals The sum of the interior angles in geodesic and translation triangles of Sl2(R)~ geometry

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5023-5036
Author(s):  
Géza Csima ◽  
Jenő Szirmai
Keyword(s):  
Universal Cover ◽  
Interior Angle ◽  
Geodesic Triangles

We study the interior angle sums of translation and geodesic triangles in the universal cover of real 2 x 2 matrices with unit determinant, as, a Thurston geometry denoted by P of SL2(R)~ geometry. We prove that the angle sum ?3i =1(?i) ? ? for translation triangles and for geodesic triangles the angle sum can be larger, equal or less than ?.

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

scholarly journals Characterisation of polyhedral products with finite generalised Postnikov decomposition

2020 ◽  
Vol 32 (5) ◽  
pp. 1253-1269
Author(s):  
Kouyemon Iriye ◽  
Daisuke Kishimoto ◽  
Ran Levi
Keyword(s):  
Finite Length ◽  
Inverse Limit ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Polyhedral Product ◽  
Postnikov Tower

AbstractA generalised Postnikov tower for a space X is a tower of principal fibrations with fibres generalised Eilenberg–MacLane spaces, whose inverse limit is weakly homotopy equivalent to X. In this paper we give a characterisation of a polyhedral product {Z_{K}(X,A)} whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include p-local and rational versions of the theorem. We end with a group theoretic application.

scholarly journals Spherical posets from commuting elements

2018 ◽  
Vol 21 (4) ◽  
pp. 593-628 ◽  
Author(s):  
Cihan Okay
Keyword(s):  
Homotopy Type ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Classifying Space ◽  
Partially Ordered ◽  
Abelian Subgroups

AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

scholarly journals The non-backtracking spectrum of the universal cover of a graph

2014 ◽  
Vol 367 (6) ◽  
pp. 4287-4318 ◽  
Author(s):  
Omer Angel ◽  
Joel Friedman ◽  
Shlomo Hoory
Keyword(s):  
Universal Cover

scholarly journals The universal cover of a quiver with relations

1983 ◽  
Vol 30 (3) ◽  
pp. 277-292 ◽  
Author(s):  
R. Martínez-Villa ◽  
J.A. De la Peña
Keyword(s):  
Universal Cover

scholarly journals A Pair of Rigid Surfaces with pg= q = 2 and K2= 8 Whose Universal Cover is Not the Bidisk

2018 ◽  
Vol 2020 (11) ◽  
pp. 3453-3493
Author(s):  
Francesco Polizzi ◽  
Carlos Rito ◽  
Xavier Roulleau
Keyword(s):  
Minimal Surfaces ◽  
Universal Cover ◽  
The Family ◽  
Albanese Map ◽  
Rigid Surfaces

Abstract We construct two complex-conjugated rigid minimal surfaces with $p_g\!=q=2$ and $K^2\!=8$ whose universal cover is not biholomorphic to the bidisk $\mathbb{H} \times \mathbb{H}$. We show that these are the unique surfaces with these invariants and Albanese map of degree 2, apart from the family of product-quotient surfaces given in [33]. This completes the classification of surfaces with $p_g=q=2, K^2=8$, and Albanese map of degree 2.

Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

2021 ◽  
Author(s):  
Jenő Szirmai
Keyword(s):  
Geodesic Sphere ◽  
Geodesic Triangle ◽  
Projective Model ◽  
Geodesic Triangles

Abstract In the present paper we study $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the ‘surface of a geodesic triangle’. Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ tetrahedron. Moreover, we generalize the famous Menelaus’s and Ceva’s theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries described by E. Molnár in [6].

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