Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space

We consider cyclic groupsG generated by an ellipto-parabolic isometry of complex hyperbolic space. We show that the Dirichlet fundamental polyhedron forG centred atz ₀ has two faces ifz ₀ is on the axis of the generator, otherwise it has infinitely many faces.     

A characterization of Fuchsian groups acting on complex hyperbolic spaces

Let G ? SU(2, 1) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is ?-Fuchsian; if G preserves a Lagrangian plane, then G is ?-Fuchsian; G is Fuchsian if G is either ?-Fuchsian or ?-Fuchsian. In this paper, we prove that if the traces of all elements in G are real, then G is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application of our main result, we show that G is conjugate to a subgroup of S(U(1)×U(1, 1)) or SO(2, 1) if each loxodromic element in G is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a ?-Fuchsian group.     

Nonlinearizable actions of dihedral groups on affine space

Let be a reductive, non-abelian, algebraic group defined over . We investigate algebraic -actions on the total spaces of non-trivial algebraic -vector bundles over -modules with great interest in the case that is a dihedral group. We construct a map classifying such actions of a dihedral group in some cases and describe the spaces of those non-linearizable actions in some examples.

     

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg ⁺↦g ⁻, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG. Partially supported by a grant from M.U.R.S.T., Italy.     

Convex polyhedra in hyperbolic space and interpolation of functions

Doklady Mathematics -     

Affine semigroups acting properly discontinuously on a hyperbolic space

In this paper we study the dynamics of properly discontinuous and crystallographic affine semigroups leaving a hyperbolic form, i.e. a quadratic from of signature (n, 1) invariant. The motivating question here is a question stated by H. Abels, G. Margulis and the author: Is the Zariski closure of a crystallographic affine semigroup leaving a hyperbolic form invariant a virtually solvable group? We proved that that for n = 2 the answer is “yes”.     

Augmentation quotients for complex representation rings of dihedral groups

Denote by D _m the dihedral group of order 2m. Let ℛ(D _m ) be its complex representation ring, and let Δ(D _m ) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δ ⁿ (D _m )/Δ ⁿ⁺¹(D _m ) for each positive integer n.     

On groups acting in phase space

The problem of the motion of a material point in a central field of general type is considered. It is shown that in the infinite-dimensional group of canonical transformations which leave the Hamiltonian function invariant there are no finite-dimensional subgroups which are significantly larger than the three-dimensional group of rotations (exact formulations in Sec. 3 and Sec. 5).Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 55–61, January, 1969.I am indebted to A. A. Kirillov for helpful discussions.     

Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut-points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well-known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.     

Norms and spectral radii of composition operators acting on the Dirichlet space

We obtain new upper bounds on the norms of univalently induced composition operators acting on the Dirichlet space and compute explicitly the norms for univalent symbols whose range is the disk minus a set of measure zero. As an application, we show that the spectral radius of every univalently induced composition operator on the Dirichlet space is equal to one.     

Affine buildings for dihedral groups

We construct 2-dimensonal thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.