An identity theorem for multi-relator groups

1991 ◽  
Vol 109 (2) ◽  
pp. 313-321 ◽  
Author(s):  
William A. Bogley
Keyword(s):  
Large Class ◽  
Finite Subgroups ◽  
Identity Theorem

In this paper, the Identity Theorem of R. C. Lyndon and the Freiheitssatz of W. Magnus are extended to a large class of multi-relator groups. Included are the two-relator groups introduced by I. L. Anshel in her thesis, where the Freiheitssatz was proved for those groups. The Identity Theorem provides cohomology computations and a classification of finite subgroups. The methods are geometric; technical tools include the original theorems of Magnus and Lyndon, as well as an amalgamation technique due to J. H. C. Whitehead.

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 Betti numbers and pseudoeffective cones in 2-Fano varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giosuè Emanuele Muratore
Keyword(s):  
Large Class ◽  
Betti Numbers ◽  
Fano Varieties ◽  
Higher Dimensional ◽  
Special Case ◽  
Answering Questions

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

scholarly journals Exceptional sequences of invertible sheaves on rational surfaces

2011 ◽  
Vol 147 (4) ◽  
pp. 1230-1280 ◽  
Author(s):  
Lutz Hille ◽  
Markus Perling
Keyword(s):  
Large Class ◽  
Complete Classification ◽  
Toric Surface ◽  
Structural Result ◽  
Toric Surfaces ◽  
Rational Surfaces ◽  
Exceptional Sequences

AbstractIn this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.

scholarly journals Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

2019 ◽  
Author(s):  
Corina Ciobotaru ◽  
Vladimir Finkelshtein ◽  
Cagri Sert
Keyword(s):  
Large Class ◽  
Closed Subgroup ◽  
Discrete Subgroup ◽  
Probability Measures ◽  
Homogeneous Dynamics ◽  
Cocompact Lattice ◽  
Geometrically Finite ◽  
Measure Rigidity ◽  
Groups Acting On Trees

Abstract We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $G$, we give a classification of the probability measures on $G/\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

An Algebraic Approach to Weakly Symmetric Finsler Spaces

2010 ◽  
Vol 62 (1) ◽  
pp. 52-73 ◽  
Author(s):  
Shaoqiang Deng
Keyword(s):  
Large Class ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
Comparison Theorems ◽  
Finsler Spaces ◽  
Volume Comparison ◽  
Algebraic Description ◽  
Interesting Class ◽  
Weakly Symmetric

AbstractIn this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. Thismeans that reversible non-Berwaldian Finsler spaces with vanishing S-curvaturemay exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.

scholarly journals Symmetric states: local unitary equivalence via stabilizers

2010 ◽  
Vol 10 (11&12) ◽  
pp. 1029-1041
Author(s):  
Curt D. Cenci ◽  
David W. Lyons ◽  
Laura M. Snyder ◽  
Scott N. Walck
Keyword(s):  
Degrees Of Freedom ◽  
Rotation Group ◽  
Equivalence Classes ◽  
Unitary Equivalence ◽  
Finite Subgroups ◽  
Local Unitary Equivalence ◽  
Symmetric States

We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group $SO(3)$, and give examples of states with discrete stabilizers.

scholarly journals Classification of Subgroups of Symplectic Groups Over Finite Fields Containing a Transvection

2016 ◽  
Vol 49 (2) ◽  
Author(s):  
S. Arias-de-Reyna ◽  
L. Dieulefait ◽  
G. Wiese
Keyword(s):  
Finite Fields ◽  
Symplectic Groups ◽  
Finite Subgroups

AbstractIn this note, we give a self-contained proof of the following classification (up to conjugation) of finite subgroups of GSp

scholarly journals The Concept of Function at the Beginning of the 20th Century: A Historiographical Approach

2018 ◽  
pp. 171
Author(s):  
Loredana Biacino
Keyword(s):  
Large Class ◽  
20Th Century ◽  
Modern Theory ◽  
Discontinuous Functions ◽  
Italian Mathematicians ◽  
Real Functions ◽  
Concept Of Function ◽  
And Function ◽  
Class Of Functions

The evolution of the concept of function at the beginning of the 20th century in France after the definitions by Dirichlet and Riemann and the introduction of several pathological functions is studied. Some young mathematicians of those years (Baire, known for his classification of discontinuous functions, Borel and Lebesgue famous for their new theories on measure and integration) made several attempts to propose a large class of functions as “accessible” objects. Their discussions, their purposes and polemics are reported often by their own words supported by a large bibliography. The contribution of some Italian mathematicians, as Vitali, is also underlined.  Some of such discussions are linked to the growth of measure and function theories, others will find mathematical answers in the modern theory of computability for real functions.

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