Subgroup Of Isometries Research Articles

Bisectors determining unique pairs of points in the bidisk

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not uniquely determine a pair of points, in the following sense: completely different sets of points share a common bisector. The above examples of this non-uniqueness are all rank [Formula: see text] symmetric spaces. However, generically, bisectors in the usual [Formula: see text] metric are such for a unique pair of points in the rank [Formula: see text] geometry [Formula: see text]. This result indicates the striking assertion that non-uniqueness of bisectors holds for “most” geometries.

A new class of monohedral pentagonal spherical tilings with GeoGebra

By a monohedral spherical tiling we mean a decomposition of the sphere by geodesic congruent polygons. Here, making use of GeoGebra, a well known free interactive mathematics software, we show how to generate new classes of monohedral non-convex triangular and new non-convex pentagonal spherical tilings, changing the side gluing rules of the regular spherical tetrahedral tiling, by means of a local action of particular subgroups of spherical isometries. In both cases each face has \pi as area measure. In relation to the new class of pentagonal tilings, we describe some of their properties and show the existence, in a special case, of an associated dihedral triangular spherical tiling, that is, a tiling composed by two sets of congruent triangles. These classes of spherical tilings have emerged as a result of an interative construction process, only possible by the use of newly produced GeoGebra tools and the dynamic interaction capabilities of this software.

Quantized mechanics of affinely rigid bodies

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configuration manifold, which is given, in our case, by the affine group or equivalently by the semi‐direct product of the linear group and the space of translations , where n equals the dimension of the “physical space.” In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2‐polar decomposition of the matrix of the internal configuration and on the Peter‐Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua.

Supersymmetric solutions of SU(2)-Fayet–Iliopoulos-gauged [formula omitted], d = 4 supergravity

We explore the construction of supersymmetric solutions of theories of N=2,d=4 supergravity with a SU(2) gauging and SU(2) Fayet–Iliopoulos terms. In these theories an SU(2) isometry subgroup of the Special-Kähler manifold is gauged together with a SU(2) R-symmetry subgroup. We construct several solutions of the CP‾3 quadratic model directly in four dimensions and of the ST[2,6] model by dimensional reduction of the solutions found by Cariglia and Mac Conamhna in N=(1,0),d=6 supergravity with the same kind of gauging. In the CP‾3 model, we construct an AdS2×S2 solution which is only 1/8 BPS and an R×H3 solutions that also preserves 1 of the 8 possible supersymmetries. We show how to use dimensional reduction as in the ungauged case to obtain Rn×Sm and also AdSn×Sm-type solutions (with different radii) in 5- and 4-dimensions from the 6-dimensional AdS3×S3 solution.

Ergodic geometry for non-elementary rank one manifolds

Let $X$ be a Hadamard manifold, and $\Gamma\subset Is(X)$ a non-elementary discrete subgroup of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to the behavior of the Poincare series of $\Gamma$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $\Gamma$-invariant conformal densities supported on the geometric limit set of $\Gamma$.

On asymptotic plateau’s problem for CMC hypersurfaces on rank 1 symmetric spaces of noncompact type

Let M n , n ≥ 3, be a Hadamard manifold with strictly negative sectional curvature KM ≤ −�, � > 0. Assume that M satisfies the strict convexity condition at infinity according to (18) (see also the definition below) and, additionally, that M admits a helicoidal one parameter subgroup {'t} of isometries (i.e. there exists a geodesic of M such that 't ((s)) = (t+s) for all s,t ∈ R). We then prove that, given a compact topological {'t}−shaped hypersurface in the asymptotic boundary @1M of M (that is, the orbits of the extended action of {'t} to @1M intersect at one and only one point), and given H ∈ R, |H| < √ �, there exists a complete properly embedded constant mean curvature (CMC) H hypersurface S of M such that @1S = . This result extends Theorem 1.8 of B. Guan and J. Spruck (11) to more general ambient spaces, as rank 1 symmetric spaces of noncompact type, and allows to be {'t}−shaped with respect to more general one parameter subgroup of isometries {'t} of the ambient space. For example, in H n , can be loxodromic −shaped, where loxodromic is a curve in S n 1 = @1H n that makes a con- stant angle with a family of circles connecting two points of S n 1 . A fundamental result used to prove our main theorem, which has interest on its own, is the extension of the interior gradient esti- mates for CMC Killing graphs proved in Theorem 1 of (7) to CMC graphs of Killing submersions.

On doubly periodic minimal surfaces in $${\mathbb {H}}^2 \times {\mathbb {R}}$$ H 2 × R with finite total curvature in the quotient space

In this paper, we develop the theory of properly immersed minimal surfaces in the quotient space \(({{\mathrm{{\mathbb {H}}^2\times {\mathbb {R}}}}})/G,\) where G is a subgroup of isometries generated by a vertical translation and a horizontal isometry (without fixed points) in \({{\mathrm{{\mathbb {H}}^2}}}\). The horizontal isometry can be either a parabolic translation along horocycles in \({{\mathrm{{\mathbb {H}}^2}}}\) or a hyperbolic translation along a geodesic in \({{\mathrm{{\mathbb {H}}^2}}}.\) We prove that if a properly immersed minimal surface in \(({{\mathrm{{\mathbb {H}}^2\times {\mathbb {R}}}}})/G\) has finite total curvature, then its total curvature is a multiple of \(2\pi \) and, moreover, we understand the geometry of the ends. The results hold true more generally for properly immersed minimal surfaces in \(M\times {\mathbb {S}}^1,\) where M is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces \(({{\mathrm{{\mathbb {H}}^2\times {\mathbb {R}}}}})/G\).

Actions without nontrivial singular orbits on Riemannian manifolds of negative curvature

We give a topological description of negatively curved Riemannian manifolds which are acted on by a closed and connected subgroup of isometries with the property that there is no nontrivial singular orbit.

The Alexandrov problem in a quotient space of ℍ2× ℝ

We prove an Alexandrov type theorem for a quotient space of $\mathbb H^2\times \mathbb R$. More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of $\mathbb H^2\times \mathbb R$ by a subgroup of isometries generated by a parabolic translation along horocycles of $\mathbb H^2$ and a vertical translation. Moreover, we construct some examples of periodic minimal surfaces in $\mathbb H^2\times\mathbb R$ and we prove a multi-valued Rado theorem for small perturbations of the helicoid in $\mathbb H^2\times\mathbb R$.

Permutation codes invariant under isometries

The symmetric group $$S_n$$ S n on $$n$$ n letters is a metric space with respect to the Hamming distance. The corresponding isometry group is well known to be isomorphic to the wreath product $$S_n \wr S_2$$ S n ? S 2 . A subset of $$S_n$$ S n is called a permutation code or a permutation array, and the largest possible size of a permutation code with minimum Hamming distance $$d$$ d is denoted by $$M(n, d)$$ M ( n , d ) . Using exhaustive search by computer on sets of orbits of isometry subgroups $$U$$ U we are able to determine serveral new lower bounds for $$M(n,d)$$ M ( n , d ) for $$n \le 22$$ n ≤ 22 . The codes are given by the group $$U$$ U and representatives of the $$U$$ U -orbits.

A geometric criterion for compactness of invariant subspaces

Let M be a non-compact homogeneous Riemannian manifold, and let Omega be a compact subgroup of isometries of M. We show, under general conditions, that the Omega-invariant subspace A (Omega) of a n ...

Isometric actions on pseudo-Riemannian nilmanifolds

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannian situation; for instance, the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with a left-invariant pseudo-Riemannian metric, we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-\(H\)-type Lie groups.

Virtual amalgamation of relatively quasiconvex subgroups

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.

On Fundamental Domains for Subgroups of Isometries Acting in

Given a fundamental polyhedron for the action of , a classical kleinian group, acting in -dimensional hyperbolic space, and , a finite index subgroup of , one obtains a fundamental domain for pasting copies of by a Schreier process. It also generalizes the side pairing generating theorem for exact or inexact polyhedra. It is proved as well that the general Möbius group acting in is transitive on “-spheres”. Hence, describing the hyperbolic -planes in the upper half space model intrinsically, and providing also an alternative proof of the transitive action on them. Some examples are given in detail, derived from the classical modular group and the Picard group.

Limit points badly approximable by horoballs

For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries Γ whose limit set is uniformly perfect, and a disjoint collection of horoballs {H j }, we show that the set of limit points badly approximable by {H j } is absolutely winning in the limit set Λ(Γ). As an application, we deduce that for a geometrically finite Kleinian group acting on $${\mathbb{H}^{n+1}}$$ , the limit points badly approximable by parabolics, denoted BA(Γ), is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that BA(Γ) has dimension equal to the critical exponent of the group. Since BA(Γ) can alternatively be described as the limit points representing bounded geodesics in the quotient $${\mathbb{H}^{n+1}/\Gamma}$$ , we recapture a result originally due to Bishop and Jones.

The group of isometries of a self-similar set

We prove that any subgroup of isometries of a Euclidean space can occur as a subgroup of isometries of a self-similar set. Furthermore the isometry group of a planar self-similar set E satisfying the open set condition must be finite, and the possible groups of isometries are restricted by the cardinality of the system of similitudes that generates E. We can prove that these results hold for self-similar sets in R3 only in the case that the generating similarities are homotheties. Lastly we prove that if a self-similar set is strongly separated, then its group of isometries must be finite.

Subgroups of isometries of Urysohn–Katětov metric spaces of uncountable density

According to Katětov (1988) [15], for every infinite cardinal m satisfying mn⩽m for all n<m, there exists a unique m-homogeneous universal metric space Um of weight m. This object generalizes the classical Urysohn universal metric space U=Uℵ0. We show that for m uncountable, the isometry group Iso(Um) with the topology of simple convergence is not a universal group of weight m: for instance, it does not contain Iso(U) as a topological subgroup. More generally, every topological subgroup of Iso(Um) having density <m and possessing the bounded orbit property (OB) is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskijʼs (1990) [35] result about the group Iso(U) being a universal Polish group.

Nonabelian Dualization of Plane Wave Backgrounds

We investigate plane-parallel wave metrics from the point of view of their (Poisson-Lie) T-dualizability. For that purpose we reconstruct the metrics as backgrounds of nonlinear sigma models on Lie groups. For construction of dual backgrounds we use Drinfel'd doubles obtained from the isometry groups of the metrics. We find dilaton fields that enable to satisfy the vanishing beta equations for the duals of the homogenous plane-parallel wave metric. Torsion potentials or B-fields, invariant w.r.t. the isometry group of Lobachevski plane waves are obtained by the Drinfel'd double construction. We show that a certain kind of plurality, different from the (atomic) Poisson-Lie T-plurality, may exist in case that metrics admit several isometry subgroups having the dimension of the Riemannian manifold. An example of that are two different backgrounds dual to the homogenous plane-parallel wave metric.

On Arnol’d’s and Kazhdan’s equidistribution problems

AbstractWe consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite; this implies the existence of an associated limiting distribution, which can be different from the underlying invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, which follows from a quantitative mean ergodic theorem for such actions. In turn, these results imply quantitative ratio ergodic theorems for isometric actions of lattices. This sheds some unexpected light on certain equidistribution problems posed by Arnol’d [Arnol’d’s Problems. Springer, Berlin, 2004] and also on the ratio equidistribution conjecture for dense subgroups of isometries formulated by Kazhdan [Uniform distribution on a plane. Tr. Mosk. Mat. Obs. 14 (1965), 299–305]. We briefly mention the general problem regarding ergodic theorems for actions of lattices on homogeneous spaces and its solution given by Gorodnik and Nevo [Duality principle and ergodic theorems, in preparation], and present a number of examples to demonstrate our results. Finally, we also prove results on quantitative equidistribution for absolutely continuous averages in transitive actions.

The hypermultiplet with Heisenberg isometry in N = 2 global and local supersymmetry

The string coupling of N=2 supersymmetric compactifications of type II string theory on a Calabi-Yau manifold belongs to the so-called universal dilaton hypermultiplet, that has four real scalars living on a quaternion-Kaehler manifold. Requiring Heisenberg symmetry, which is a maximal subgroup of perturbative isometries, reduces the possible manifolds to a one-parameter family that describes the tree-level effective action deformed by the only possible perturbative correction arising at one-loop level. A similar argument can be made at the level of global supersymmetry where the scalar manifold is hyper-Kaehler. In this work, the connection between global and local supersymmetry is explicitly constructed, providing a non-trivial gravity decoupled limit of type II strings already in perturbation theory.