Quasi-isometric rigidity of Fuchsian buildings

We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. Together with the results of Bourdon-Pajot and Kleiner, it implies the quasi-isometric rigidity for Fuchsian buildings: any quasi-isometry between two Fuchsian buildings that admit cocompact lattices must lie at a finite distance from an isomorphism.     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, [(Z)\tilde]\tilde{Z} , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

A Spectral Correspondence for Maaß Waveforms

Let be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order in an indefinite quaternion division algebra over . Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maa? newforms for the Hecke congruence group , when d is the discriminant of the maximal order . We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for and the newforms of level d, respectively. Submitted: July 1998, final version: January 1999.     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

Free Algebras of Automorphic Forms on the Upper Half-Plane

We find all pairs (,a) consisting of a cocompact Fuchsian group of genus zero and an automorphy factor a of for which the graded algebra of a--automorphic forms is free.     

Computations of K- and L-Theory of Cocompact Planar Groups

The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic K- and L-theory and the topological K-theory of cocompact planar groups (=cocompact N.E.C-groups) and of groups G appearing in an extension where is a finite group and the conjugation -action on ⁿ is free outside . These computations apply, for instance, to two-dimensional crystallographic groups and cocompact Fuchsian groups.     

Geometric quantization for proper actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.     

Bifurcation of Hyperbolic Planforms

Motivated by a model for the perception of textures by the visual cortex in primates, we analyze the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D\mathcal {D} (Poincaré disc). We make use of the concept of a periodic lattice in D\mathcal {D} to further reduce the problem to one on a compact Riemann surface D/\varGamma\mathcal {D}/\varGamma, where Γ is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows us to use the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called “H-planforms”, by analogy with the “planforms” introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are, however, not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.     

Relativity of the spectrum and discrete groups on hyperbolic spaces

We give a simple proof of the analytic continuation of the resolvent kernel for a convex cocompact Kleinian group.

     

Solvgroups are not almost convex

We show that no cocompact discrete group based on solvgeometry, Sol, is almost convex. This reflects the geometry of Sol, and implies that the Cayley graph of a cocompact discrete group based on Sol cannot be efficiently constructed by finitely many local replacement rules.This research was supported in part by NSF Research Grants.     

On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

We give a review of the modern theory of isomonodromic deformations of Fuchsian systems discussing both classical and modern results, such as a general form of the isomonodromic deformations of Fuchsian systems, their differences from the classical Schlesinger deformations, the Fuchsian system moduli space structure and the geometric meaning of new degrees of freedom appeared in a non-Schlesinger case. Using this we illustrate some general relations between such concepts as integrability, isomonodromy and Painlevé property. The work is supported by N.Sh.-6849.2006.1 and RFBR 07-01-00526 grants.     

Fuchsian convex bodies: basics of Brunn–Minkowski theory

The hyperbolic space ${\mathbb{H}^d}$ can be defined as a pseudo-sphere in the (d + 1) Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that ${\mathbb{H}^d/\Gamma}$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov–Fenchel and Brunn–Minkowski inequalities. Here the inequalities are reversed.     

Shadows Of Mapping Class Groups: Capturing Convex Cocompactness

We strengthen the analogy between convex cocompact Kleinian groups and convex cocompact subgroups of the mapping class group of a surface in the sense of B. Farb and L. Mosher. Received: August 2006, Accepted: January 2007, Revision: February 2007     

Inclusion relations between the Bers embeddings of Teichmüller spaces

We prove that if the Bers embeddings of the Teichmüller spaces of infinitely generated Fuchsian groups are coincident, then these Fuchsian groups are the same.     

SL(n,C)中的一些特殊可解子群及应用

给出了SL(n,C)中的几类特殊可解子群,并应用于Fuchs系统.由Fuchs方程的单值群的可解性与其可积性的关系,给出了其可积的一些条件.     

New extension phenomena for solutions of tangential Cauchy–Riemann equations

In our recent work, we showed that C^∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ?² are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.

In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.

Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.     

On globally nilpotent differential equations

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence.This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems.     

On Cyclic Groups of Automorphisms of Riemann Surfaces

The question of extendability of the action of a cyclic groupof automorphisms of a compact Riemann surface is considered.Particular attention is paid to those cases corresponding toSingerman's list of Fuchsian groups which are not finitely-maximal,and more generally to cases involving a Fuchsian triangle group.The results provide partial answers to the question of whichcyclic groups are the full automorphism group of some Riemannsurface of given genus g>1.     

Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.