Search results for "group theory"
showing 10 items of 703 documents
On Radon transforms on compact Lie groups
2016
We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.
Regularity properties of spheres in homogeneous groups
2015
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an alg…
Counting and equidistribution in quaternionic Heisenberg groups
2020
AbstractWe develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.
Regularity of sets with constant horizontal normal in the Engel group
2012
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic horizontal upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the intrinsic horizontal upper-graph of a function t…
Metric equivalences of Heintze groups and applications to classifications in low dimension
2021
We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups. Then we will see how these results together with the existing tools related to isometries can be applied to groups of dimension 4 and 5 in particular. Thus we take steps towards determining all the equivalence classes of groups up to isometry and quasi-isometry. We completely solve the classification up to isometry for simply connected solvable groups in dimension 4, and for the subclass of groups of polynomial growth in dimension 5.
Rigidité, comptage et équidistribution de chaînes de Cartan quaternioniques
2020
We prove an analog of Cartan's theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.; Nous montrons un analogue d'un théorème de Cartan, disant que les transformations préservant les chaînes sur le bord d'un espace hyperbolique quaternionien est une transformation projective. Nous donnons un résultat de comptage et d'équidistribution pour une orbite de chaînes arithmétiques dans le groupe de Heisenberg quaternionique.
On the non-triviality of the torsion subgroup of the abelianized Johnson kernel
2022
The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3-manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2-dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the …
Generalized Dehn twists in low-dimensional topology
2021
The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description in terms of decorated trees and the Hopf-algebraic framework underlying their construction. Going t…
Sets versus trial sequences, Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920
2010
Abstract The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix ) of two …
Coprime actions and correspondences of Brauer characters
2017
We prove several results giving substantial evidence in support of the conjectural existence of a Glauberman–Isaacs bijection for Brauer characters under a coprime action. We also discuss related bijections for the McKay conjecture.