Fundamental Polygon Research Articles

Generic fundamental polygons for surfaces of genus three

Generic fundamental polygons for surfaces of genus three

THE CULLER-SHALEN SEMINORMS OF THE (-2, 3, n) PRETZEL KNOT

We show that the [Formula: see text]-character variety of the (-2, 3, n) pretzel knot consists of two (respectively three) algebraic curves when 3 ∤ n (respectively 3 | n) and given an explicit calculation of the Culler-Shalem seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgerise yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

On Lacunary Analogs of the Poincare Theta-Series and Their Applications

We consider Fuchsian groups of linear-fractional transformations such that each vertex of the fundamental polygon is common for an even or infinite number of fundamental congruent polygons meeting at this point. The whole collection of transformations splits into two disjoint sets. For these sets we introduce two lacunary kernels whose sum represents the well-known analog of Chibrikova and Sil'vestrov's kernel and study their properties. We introduce automorphic forms of dimension −4m that differ from the Poincare theta-series. We indicate an application of one of the constructed lacunary kernels which does not include the Cauchy kernel to solving some boundary value problem with a shift of the contour inside the domain.

Calculs de Présentations de Groupes Fuchsiens via les Graphes Rubanés

We show how to compute a geometrical presentation for a finitely generated fuchsian group G from the explicit knowledge of a fundamental polygon for G. We use a previous work on fat-graphs [4]. In particular, our result is a complement to the works of R.S. Kulkarni [10] and S. Johansson [5] on arithmetic fuchsian groups.

Gluing Affine 2-Manifolds with Polygons

Affine structures on surfaces are constructed by gluing polygons. The geometry of the affine surface depends on the shape of the polygon(s) and the particular gluing transformations used. The affine version of the Poincare fundamental polygon theorem expresses the fundamental group and holonomy of the surface in terms of the gluing data. The theorem may be used to construct all complete affine structures on the 2-torus. The space of inequivalent holonomy representations of such structures is homeomorphic to R2.

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between π/p and π/q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.

Drawing Graphs on Surfaces

Every graph that is 2-cell embedded in a closed orientable surface can be drawn in some fundamental polygon of the surface so that the boundary of the polygon consists of edges and vertices of the graph; it can also be drawn so that all the vertices of the graph are inside the polygon and no edges cross the boundary of the polygon more than once. A necessary and sufficient condition is found to determine whether or not a given embedding of a graph in a surface has one or the other representation in the standard polygon of the surface, the one of the form $\{a_{1}B_{1}a_{1}^{-1}b_{1}^{-1}a_{2}b_{2}a_{2}^{-1}b_{2}^{-1}....$ This leads to a polynomial-time algorithm, assuming that the surface is fixed.

Symmetries of Planar Growth Functions. II

Let $G$ be a finitely generated group, and let $\Sigma$ be a finite generating set of $G$. The growth function of $(G,\Sigma )$ is the generating function $f(z) = \sum \nolimits _{n = 0}^\infty {{a_n}{z^n}}$, where ${a_n}$ is the number of elements of $G$ with word length $n$ in $\Sigma$. Suppose that $G$ is a cocompact group of isometries of Euclidean space ${\mathbb {E}^2}$ or hyperbolic space ${\mathbb {H}^2}$, and that $D$ is a fundamental polygon for the action of $G$. The full geometric generating set for $(G,D)$ is $\{ g \in G:g \ne 1$ and $gD \cap D \ne \emptyset \}$. In this paper the recursive structure for the growth function of $(G,\Sigma )$ is computed, and it is proved that the growth function $f$ is reciprocal $(f(z) = f(1/z))$ except for some exceptional cases when $D$ has three, four, or five sides.

Cuntz—Krieger algebras associated with Fuchsian groups

AbstractFor Fuchsian groups of the first kind containing parabolic elements, it is shown that the action on a suitable disconnection of the limit circle generates a Cuntz—Krieger C*-algebra. This clarifies and generalizes the situation of the subalgebra within O2, and provides a new proof of the simplicity and nuclearity of certain Cuntz—Krieger algebras. The proof relies on the Markov partition obtained from a suitable fundamental polygon for the group. Counter examples are given if an unsuitable fundamental polygon is used.

On symmetric circular polygons

Consider a polygon in the Riemann sphere with edges formed by 4g circular arcs, identified pairwise in a canonical fashion via Möbius transformations to define a compact surface of genus g. It is shown that if both the polygon and the generators enjoy a rotational symmetry of order k, where k > 1 is a factor of g, if the interior angles total 2π, and if a certain additional normalization is satisfied, then the product of the commutators of these transformations is the identity. This provides easily computable examples of continuous families of projective structures for any g > 1, each with a common fundamental polygon. AMS No. 30F10, 30F35

Symmetries of planar growth functions. II

Let G G be a finitely generated group, and let Σ \Sigma be a finite generating set of G G . The growth function of ( G , Σ ) (G,\Sigma ) is the generating function f ( z ) = ∑ n = 0 ∞ a n z n f(z) = \sum \nolimits _{n = 0}^\infty {{a_n}{z^n}} , where a n {a_n} is the number of elements of G G with word length n n in Σ \Sigma . Suppose that G G is a cocompact group of isometries of Euclidean space E 2 {\mathbb {E}^2} or hyperbolic space H 2 {\mathbb {H}^2} , and that D D is a fundamental polygon for the action of G G . The full geometric generating set for ( G , D ) (G,D) is { g ∈ G : g ≠ 1 \{ g \in G:g \ne 1 and g D ∩ D ≠ ∅ } gD \cap D \ne \emptyset \} . In this paper the recursive structure for the growth function of ( G , Σ ) (G,\Sigma ) is computed, and it is proved that the growth function f f is reciprocal ( f ( z ) = f ( 1 / z ) ) (f(z) = f(1/z)) except for some exceptional cases when D D has three, four, or five sides.

Trace moduli for quasifuchsian groups

O . Introduction. In earlier work [13, 14, 15] we consider the problem of finding moduli for the Teichmtiller space of marked Fuchsian groups, T ( G ) . Our approach is to decompose the Fuchsian group G into two-generator subgroups. We construct fundamental polygons for these subgroups and then combine them into a single fundamental polygon P for the full group. Although the number of sides P has is bigger than the minimum number of sides a polygon must have, it is not much bigger. Geometric moduli are determined as lengths of certain of the sides of P, and distances between particular pairs of the other sides. These geometric moduli also have an interpretation as the traces of a particular set S of elements G . This parametrization determines a real analytic equivalence between T(G) and a simply connected subset of Euclidean space R P , for appropriate p . In this paper, we generalize these results to a special class of quasifuchsian groups. In the space of marked quasifuchsian groups, stability considerations tell us that the same set of traces can be used as moduli in a neighborhood of the Fuchsian locus. The number of faces of the fundamental polyhedra in H 3 for these groups, however, can be arbitrarily large. O ur class contains quasifuchsian groups that have fundamental polyhedra with a constant small number of sides and no interior vertices, and can be parametrized by this same set of traces. It extends out of the neighborhood of the Fuchsian locus and intersects the boundary in B-groups.

Generic Dirichlet polygons and the modular group

The concept of “marked polygon”, made explicit in this paper, is implicit in all studies of the relationships between the edges and vertices of a fundamental polygon for Fuchsian group, as well as in the topology of surfaces. Once the matching of the edges under the action of the group is known, one can deduce purely combinatorially the distribution of the vertices into equivalence classes, or cycles. Knowing a little more, the order of the rotation group fixing a vertex in each cycle, we can write down a presentation for the group.

On lifting Fuchsian Groups

Let Γ be a discrete subgroup of G = PSL(2,ℝ) and p be the canonical map from the universal covering group on to PSL(2,ℝ). By a continuity argument on a fundamental polygon for Γ acting on the hyperbolic plane 2, Milnor (4) obtained a presentation for p−1(Γ) whenever 2/Γ is compact and of genus zero. Using Teichmüller theory and a double Reidemeister-Schreier process, Macbeath(2) showed that the general compact case can be deduced from Milnor's result. We give here a method of obtaining a presentation which uses only the geometry of a Dirichlet region and which applies equally to the non-compact case.

On Poincaré's theorem for fundamental polygons

On Poincaré's theorem for fundamental polygons

On Torsion-Free Discrete Subgroups of PSL(2, C) with Compact Orbit Space

In 1883 Poincaré [13] recognized that the discrete subgroups of PSL(2, C) could be extended from their natural action on the complex plane to acting on hyperbolic 3-space and he attempted to analyze these groups in an analogous manner to his classical treatment of Fuchsian groups, with fundamental polyhedra playing the role of the fundamental polygons for Fuchsian groups. This approach, however, did not lead very far, perhaps not surprisingly when one appreciates the close connection between the geometry of these groups and the topology of 3-manifolds. Since that time the state of knowledge remained essentially unchanged until 1964 when work by Ahlfors [1] and soon afterwards by Bers [3] revitalized the subject of Kleinian groups. The modern approach tends to use analytic methods, although recently Marden [11] has had considerable success in carrying forward Poincaré's geometric approach.

Fundamental polygons for Fuchsian groups

Fundamental polygons for Fuchsian groups

Canonical polygons for finitely generated Fuchsian groups

We consider finitely generated Fuchsian groups G. For such groups Fricke defined a class of fundamental polygons which he called canonical. The two most important distinguishing properties of these polygons are that they are strictly convex and have the smallest possible number of sides. A canonical polygon P in Fricke's sense depends on a choice of a certain standard system of generators S for G. Fricke proved that for a given G and S canonical polygons always exist. His proof is rather complicated. Also, Fricke's polygons are not canonical in the technical sense; there are infinitely many P for a given G and S. In this paper, we shall construct a uniquely determined fundamental polygon P which satisfies all of Fricke's conditions for every given G of positive genus and for every given S. We call this P a canonical Fricke polygon. For every given G of genus zero, and S, we shall define a uniquely determined fundamental polygon P which we call a canonical polygon without accidental vertices. From this P, one can obtain in infinitely many ways polygons satisfying l~'ricke's conditions. Our canonical polygons are invariant under similarity transformations of the group G if G is of the first kind. If G is of the second kind, this statement remains true after a suitable modification which will be clear from the construction. The proof involves elementary explicit constructions, and continuity arguments which use quasiconformal mappings, as developed by Ahlfors and Bers. We give a geometric interpretation of the canonical polygons in the last section. This interpretation provided the heuristic idea for the formulation of the main theorem.

Hyperbolic generators for Fuchsian groups

The standard system of generators for a Fuchsian group contains, in general, some elliptic and parabolic transformations. It is not, however, well known that one can always choose a system of generators consisting only of hyperbolic transformations.2 We will obtain this result by establishing a stronger statement: there exists a convex noneuclidean fundamental polygon, whose sides are identified by hyperbolic transformations. Clearly, fundamental other than the Fricke polygons are needed. Such are considered in [6] where most of the facts which we shall need are developed. We will often omit details and proofs when they would be only slight modifications of those in [6]. 1. Preliminaries. A finitely generated discontinuous group G of conformal self-mappings of the unit disc U is called a Fuchsian group. Let ir denote the canonical mapping of U onto U/G. Then U/G, with the induced conformal structure, is a Riemann surface. Given integers g, h, n, I and P1, * * *, v. satisfying

Fundamental polygons of fuchsian and fuchsoid groups

Fundamental polygons of fuchsian and fuchsoid groups