Fundamental domains for planar discontinuous groups and uniform tilings

In this paper, we classify combinatorially different fundamental domains for any given planar discontinuous group and we give an algorithm for the complete enumeration of uniform tilings of any complete, simply connected, two-dimensional Riemannian manifold of constant curvature.Supported by Hungarian National Foundation for Scientific Research, Grant No. 1238/86.     

A properly discontinuous free group of affine transformations

In this paper we prove that a fully irreducible outer automorphism relative to a non-exceptional free factor system acts loxodromically on the relative free factor complex as defined in Handel and Mosher (Relative free splitting and relative free factor complexes I: hyperbolicity, 2014. arXiv:1407.3508v1). We also prove a north-south dynamic result for the action of such outer automorphisms on the closure of relative outer space.     

On the properly discontinuous subgroups of affine motions

The fundamental groupΓ of a compact complete affine manifold is represented as an affine crystallographic subgroup of Aff(n). L.S.Auslander conjectured thatΓ is virtually solvable. Our purpose is to find the algebraic condition onΓ which leads affirmative answer to the conjecture.     

Combinatorial classification of fundamental domains of finite area for planar discontinuous isometry groups

Supported by the Hung. Nat. Found. Sci. Res. Grant No. 1238/86.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

Dual affine quantum groups

Let be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let be the dual Lie bialgebra. By dualizing the quantum double construction – via formal Hopf algebras – we construct a new quantum group , dual of . Studying its specializations at roots of 1 (in particular, its semi-classical limits), we prove that it yields quantizations of and (the formal Poisson group attached to ), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type. Received January 27, 1999     

Properly discontinuous group actions on affine homogeneous spaces

Let G be a real algebraic group, H ≤ G an algebraic subgroup containing a maximal reductive subgroup of G, and Γ a subgroup of G acting on G/H by left translations. We conjecture that Γ is virtually solvable provided its action on G/H is properly discontinuous and ΓG/H is compact, and we confirm this conjecture when G does not contain simple algebraic subgroups of rank ≥2. If the action of Γ on G/H (which is isomorphic to an affine linear space Aⁿ) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for n ≤ 5.     

Convex affine domains and Markus conjecture

In this paper we show that any properly convex quasi-homogeneous affine domain with irreducible projective automorphism group is projectively equivalent to a homogeneous affine domain. Then as an application we answer positively the Markus conjecture when the manifold M is in a certain class of convex affine manifolds.Mathematics Subject Classification (1991): 51M10, 57S25This work was supported by Korea Research Foundation Grant (KRF-2002-070-C00010).in final form: 6 October 2003     

Cohomology of discontinuous groups

Prasad (1979) proved that the set of all equivalence classes of representationsp of a Fuchsian group Γ whose restrictions to the cyclic subgroups Γ _i -(c _i ) corresponding to the parabolic and elliptic elements of Γ occurring in the structure of Γ, are given, is a complex analytic manifold. In the process the author has proved thatH ¹(X,A)≈P ¹(Γ,ρ) and with suitable notation. This paper gives the corresponding results to the two above mentioned results, when in place of Γ we consider any discontinuous group of Poincare isometries Δ, and when similar assumptions are made.     

On properly essential classical conformal diffeomorphism groups

In this article, we prove that various classical conformal diffeomorphism groups, which are known to be essential (Banyaga, J Geom 68(1–2):10–15, 2000), are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological equation. Furthermore, we study the orbit of a tensor field under the action of the conformal diffeomorphism group for these classical conformal structures. On every closed contact manifold, we find conformal contact forms that are not diffeomorphic.     

Amalgamated products and properly 3-realizable groups

In this paper, we show that the class of all properly 3-realizable groups is closed under amalgamated free products (and HNN-extensions) over finite groups. We recall that G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π₁(K)≅G and whose universal cover has the proper homotopy type of a 3-manifold (with boundary).     

A presentation for reduced extended affine weyl groups

In 1985 K. Saito [Sal] introduced the concept of an extended affine Weyl group (EAWG), the Weyl group of an extended affine root system (EARS). In [A2, Section 5J, we gave a presentation called “a presentation by conjugation” for the class of EAWGs of index zero, a subclass of EAWGs. In this paper we will give a presentation wh.ich we call a “generalized present.ation by conjugation” for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to “a presentation by conjugation”. Our main result states that when the nullity of the EARS is 2, these two presentations coincide that is, EAWGs of nullity 2 have “a presentation by conjugation”. In [ST] another presentation for EAWGs of nullity 2 is given.