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The classification of discrete 2-generator subgroups of PSL(2, R )
Abstract
This paper gives a short geometric algorithm for deciding the discreteness of most 2-generator subgroups of PSL(2,R), as well as a self-contained algorithmic approach to the complete classification.
... The proof of Theorem 1.1 relies heavily on the description of all generating pairs of two-generated Fuchsian groups given by Fine-Knapp-Matelsky-Purzitsky-Rosenberger [2,4,9,11,13]. ...
... Throughout this section O will denote the orbifold S 2 (p 1 , p 2 , p 3 ) and G its fundamental group. In [9] (see also Knapp [4] for the particular case of generating pairs consisting of elliptic elements and [2] for a more general discussion) Matelski implicitly shows that if T is a generating pair of G (the arguments in [9] work also for non-hyperbolic triangle groups) then T is equivalent to T where one of the following holds: ...
... Throughout this section O will denote the orbifold S 2 (p 1 , p 2 , p 3 ) and G its fundamental group. In [9] (see also Knapp [4] for the particular case of generating pairs consisting of elliptic elements and [2] for a more general discussion) Matelski implicitly shows that if T is a generating pair of G (the arguments in [9] work also for non-hyperbolic triangle groups) then T is equivalent to T where one of the following holds: ...
We extend the result of \cite{Dutra} to generating pairs of triangle groups, that is, we show that any generating pair of a triangle group is represented by a special almost orbifold covering.
... It was proved by Jørgensen [12] that Γ ≤ PSL (2, C) is discrete if and only if every nonelementary twogenerator subgroup of Γ is discrete. Accordingly, significant progress in the literature has occurred since then towards a resolution of the discreteness problem for subgroups of PSL (2, C) through the examination of twogenerator subgroups (see [7], [11], [10], [14], [17] and the references therein). A particularly remarkable result was presented by Gilman in [8] with an algorithm for deciding the discreteness of the subgroups of PSL (2, R) . ...
... Assume that f 2 (x * ) < α * . Consider the vector ⃗ v 1 ∈ T x * ∆ 27 with the following coordinates: 3,4,5,7,9,10,11,13,14,15,17,18,19,21 ...
... This is a contradiction. Hence, we find that 2,5,6,7,8,9,12,13,14,15,16,19,20 For l ∈ J , i ∈ K ′ = {2, 6, 16, 20}, j ∈ L ′ = {8, 12, 22, 26}, and k ∈ N , we calculate ...
Let ξ and η be two noncommuting isometries of the hyperbolic 3-space ³ so that Γ = 〈ξ η〉 is a purely loxodromic free Kleinian group. For γ ∈ Γ and z ∈ ³ , let d γ z denote the hyperbolic distance between z and γ(z) . Let z 1 and z 2 be the midpoints of the shortest geodesic segments connecting the axis of ξ to the axes of ηξη ⁻¹ and η ⁻¹ ξη, respectively. In this manuscript, it is proved that if d γz 2 < 1:6068::: for every γ ∈[η, ξ ⁻¹ ηξ, ξηξ ⁻¹ ] and d ηξη -1 z 2 ≤ d ηξη -1 z 1 , then |trace ² (ξ) - 4| + |trace(ξηξ ⁻¹ η ⁻¹ ) - 2| ≥ 2 sinh ² (1/4 log α) = 1:5937:::: Above α = 24:8692::: is the unique real root of the polynomial 21x ⁴ - 496x ³ - 654x ² + 24x + 81 that is greater than 9. Generalizations of this inequality for finitely generated purely loxodromic free Kleinian groups are also proposed.
... In addition to providing inequalities which give necessary and sufficient conditions for discreteness, this work provides a conceptual basis for constructing examples. In a subsequent paper we show that the inequalities imply J0rgensen's inequality in the hyperbolic case and use the conceptual framework to construct counterexamples to results claimed in [9]. ...
... Remark. Using the results of [9] and [20], one could improve Propositions 8.4 and 8.5 to give necessary and sufficient conditions for (A ln , B) to be discrete and not discrete free. It is not clear whether these conditions would be clean and useful enough. ...
... It is not clear whether these conditions would be clean and useful enough. This will be discussed in a forthcoming paper which will also include a counterexample to case (6) of results claimed in [9]. ...
Conditions for a subgroup, F , of PSL (2, R ) to be discrete have been investigated by a number of authors. Jørgensen's inequality [ 5 ] gives an elegant necessary condition for discreteness for subgroups of PSL (2, C ). Purzitsky, Rosenberger, Matelski, Knapp, and Van Vleck, among others [ 12, 13, 14, 9, 16, 17, 18, 19, 20, 7, 21 ] studied two generator discrete subgroups of PSL (2, R ) in a long series of papers. The complete classification of two generator subgroups was surprisingly complicated and elusive. The most complete result appears in [20] .
In this paper we use the results of [20] to prove that a nonelementary subgroup F of PSL (2, R ) is discrete if and only if every non-elementary subgroup, G , generated by two hyperbolics is discrete (Theorem 5.2) and that F contains no elliptics if and only if each such G is free (Theorem 5.1). Thus, we produce necessary and sufficient conditions for a non-elementary subgroup F of PSL (2, R ) to be a discrete group without elliptic elements (Theorem 6.1) or a discrete group containing only hyperbolic elements (Theorem 7.1).
... Consider the numbering of faces of P(α, β) as shown in its projection in Figure 3. Let ρ(j, k) be the hyperbolic distance between the faces j and k. Then we write A = cosh α = cosh ρ (3,4), B = cosh β = cosh ρ (7,8), u = cosh ρ(1, 7) = cosh ρ (2,8), v = cosh ρ(3, 6) = cosh ρ (4,5). ...
... Consider the numbering of faces of P(α, β) as shown in its projection in Figure 3. Let ρ(j, k) be the hyperbolic distance between the faces j and k. Then we write A = cosh α = cosh ρ (3,4), B = cosh β = cosh ρ (7,8), u = cosh ρ(1, 7) = cosh ρ (2,8), v = cosh ρ(3, 6) = cosh ρ (4,5). ...
... Denote by G α,β the Gram matrix of the polyhedron P(α, β): (4,5,6,7,8), respectively, we will get following four equations. ...
We consider two families of hyperbolic polyhedra. With one set of face pairings, these polyhedra give the convex core of certain quasiFuchsian punctured torus groups. With additional face pairings, they are related to hyperbolic cone manifolds with singularities over certain links. For both families we derive formulae relating the dihedral angles, side lengths and the volume of the polyhedron.
... It turns out that Γ = f, g has an invariant plane in one of the following cases: (1) γ < 0 and β ≤ −4; (2) γ > 0 and β ≥ −4. Such discrete groups were investigated, for example, in [13] and [8,14,15], respectively. If γ < 0 and β > −4, then Γ is truly spatial (non-elementary and without invariant plane) and this case is treated in [11]. ...
... Since Γ is discrete, H must be discrete. By [15] or [2], the group H is discrete if and only if either ...
... The subgroup H = e, f gf −1 of Γ keeps the plane κ invariant and is conjugate to a subgroup of PSL(2, R). By [15], H is discrete if and only if h 2 = ef gf −1 is either a hyperbolic, or parabolic, or primitive elliptic element of order p ≥ 3. ...
We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain two-dimensional slice through this space. In order to do this we gather together known discreteness criteria for two-generator groups and present them in the form of conditions on parameters. We complete the description by giving discreteness criteria for groups generated by a parabolic and a -loxodromic elements whose commutator has real trace and present all orbifolds uniformized by such groups.
... We will also assume that p 1 ≤ p 2 ≤ p 3 . We need the following proposition which was independently proved (at least implicitly) by Fine-Gilman-Knapp-Maskit-Matelsky-Purzitsky-Rosenberger [3,5,8,16,21,23]. We will follow the argument given by Gilman-Maskit-Matelski [5,16], see also [4]. ...
... We need the following proposition which was independently proved (at least implicitly) by Fine-Gilman-Knapp-Maskit-Matelsky-Purzitsky-Rosenberger [3,5,8,16,21,23]. We will follow the argument given by Gilman-Maskit-Matelski [5,16], see also [4]. ...
We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking.
... For groups G α generated by reflections in the sides of triangles and quadrilaterals we find all Coxeter groups α , such that G α is a subgroup of α ; the indices of these subgroups can be easily obtained from the figures of Section 5. Decompositions of triangles obtained by reflections of Coxeter triangles were already studied . The spherical case is analysed in [1], the hyperbolic case is analysed for special types of triangles in [2], [3] and [4]. The paper [5] contains the solution for hyperbolic triangles having a non-fundamental angle and the complete list of Coxeter decompositions of hyperbolic triangles; a list of decompositions of the hyperbolic quadrilaterals of some special type is also obtained there. ...
... On the other hand, analogously to case (1), we have N ≥ 2k + 2 (seeFigure 3(a) again).2, 3, 7) it is easy to see that there is no such decomposition. ...
LetPbe a polygon on hyperbolic plane H2. A Coxeter decomposition of a polygonPis a non-trivial decomposition ofPinto finitely many Coxeter polygonsFi, such that any two polygonsF1andF2having a common side are symmetric with respect to this common side. In this paper we classify the Coxeter decompositions of triangles, quadrilaterals and ideal polygons.
... (c) Suppose that ∆ is a hyperbolic triangle with angles π/2, π/n, kπ/q. From the list of all hyperbolic triangles with one non-primitive and two primitive angles [16], we have that k = 2 and q = n ≥ 7 is odd. ...
... Since q ≥ 7, the link of the vertex made by ζ, σ, and τ is also a hyperbolic triangle and again from [16] we conclude that m = q = n. ...
We give a complete list of orbifolds uniformised by discrete non-elementary two-generator subgroups of PSL(2,C) without invariant plane whose generators and their commutator have real traces.
... If such a polygon is a triangle with two primitive and one non-primitive angle, we use Figure 2 that gives a list of all such triangles in hyperbolic and spherical cases which 'generate' a discrete reflection group. The list of the hyperbolic triangles first appeared in Knapp's paper [21], then Matelski gave a nice geometric proof [23]. The list for the spherical case can be found in [5]. ...
... Clearly, there are no such discrete groups (cf. [23]). ...
In this paper we give necessary and sufficient conditions for discreteness of a group generated by a hyperbolic element and an elliptic one of odd order. This completes the classification of discrete groups with non--loxodromic generators in the class of two-generator groups with real parameters. The criterion is given also as a list of all parameters that correspond to discrete groups. An interesting corollary of the result is that the group of the minimal known volume hyperbolic orbifold has real parameters.
... Coxeter decompositions of hyperbolic triangles were studied in [10], [11], [12], [13] and [2]. Coxeter decompositions of hyperbolic tetrahedra are listed in [4]. ...
Let X be a space of constant curvature and P be a convex polyhedron in X. A Coxeter decomposition of the polyhedron P is a decomposition of P into finitely many Coxeter polyhedra, such that any two polyhedra having a common facet are symmetric with respect to this facet. In this paper we classify Coxeter decompositions of simplices in hyperbolic space of dimension greater than 3. The problem is close to the classification of the finite index subgroups in the discrete hyperbolic reflection groups.
... The problem of deciding discreteness of a finitely generated linear group has been explored in various settings. The generating pairs of 2-generated Fuchsian groups were characterised in a series of papers by Rosenberger, Kern-Isberner, Purzitsky and Matelski [14,17,21,23,24,25,26,27,28,30]. This characterisation takes the form of an algorithm, in which a core procedure is the repeated application of Nielsen transformations to the generators that minimise their traces. ...
We provide algorithms to decide whether a finitely generated subgroup of is discrete, solve the constructive membership problem for finitely generated discrete subgroups of , and compute a fundamental domain for a finitely generated Fuchsian group. These algorithms have been implemented in Magma for groups defined over real algebraic number fields.
... These lemmata may be proved with a direct computation. Here we offer the following proof by Matelski [14] which is more elegant and revealing. The first arguments of the proof of 2.10 are the same of the proofs of 2.11 and 2.12, hence we may merge the proofs of these lemmata in a unique one and then discussing case by case. ...
Let S be a closed surface of genus g. In this paper, we investigate the relationship between hyperbolic cone-structure on S and representations of the fundamental group into PSL. We prove that, if S has genus 2, then every representation with Euler number sends a simple non-separating curve to an elliptic element. After that, we use this result to derive that every representation with arises as the holonomy of some hyperbolic cone-structure. Finally we consider surfaces of genus greater than 2 and we show that, under suitable condition, every representation PSL with Euler number arises as holonomy of a hyperbolic cone-structure on S with a single cone point of angle .
... Cette proposition est l'outil de base de la classification des sous-groupesà deux générateurs de PSL(2,R)(voir [Mat82,Gil95] ...
The main theme in this work is the study of representations of
surface groups in the real non-compact Lie group PU(2,1), which is the isometry
group of the complex hyperbolic plane. More precisely, we are
interested in the representations of the fundamental group of the once
punctured torus in PU(2,1). Our main
result states the existence of a 3 parameter family of discrete,
faithful an type preserving representations of the punctured torus
fundamental group in PU(2,1). The main tool in the proof is a new
type of hypersurface of the complex hyperbolic plane, which we use to build
fondamental domains. We also study the decomposability of
representations, and give criteria to guaranty the existence of a
decomposition, both in terms of traces and cross ratios.
... We observe that Ax [A,B] is disjoint from the axes Ax A and Ax B as follows. Following [14] construct h, the perpendicular from p B to Ax A . By the trace minimality of the stopping generators, h intersects Ax A between p A and A −1 (p) (or vice-versa) [5]. ...
In this chapter, we study conformal maps between domains in the extended complex plane ; these are one-to-one meromorphic functions. Our goal here is to characterize all simply connected domains in the extended complex plane. The first two sections of this chapter study the action of a quotient of the group of two-by-two nonsingular complex matrices on the extended complex plane, namely, the group PSL(2, ℂ) and the projective special linear group. This group is also known as the Möbius group. In the third section we characterize simply connected proper domains in the complex plane by establishing the Riemann mapping theorem (RMT). This extraordi- nary theorem tells us that there are conformal maps between any two such domains.
... A geometrically motivated algorithm for deciding whether or not a twogenerator real matrix group is discrete was described by Gilman-Maskit [4]. The geometric form of the algorithm built upon algebraic forms developed by Purzitsky and Rosenberger [15] and [16] and an incomplete approach due to Matelski [14]. ...
There are certain sequences of words in the generators of a two- generator subgroup ofSL(2;C) that frequently arise in the Teichmuller theory of hyperbolic three-manifolds and Kleinian groups. In this pa- per we establish the connection between two such families, the family of Farey words that have been used by Keen-Series to understand the boundaries of the Quasifuchsian space of surfaces of type (1; 1) and the Schottky space of surfaces of type (0; 4) and the sequences of words that arise in discreteness algorithms for two-generator subgroups of SL(2;C) studied by Rosenberger-Purzitsky and Gilman-Maskit and Jiang.
... Thus we need a result that tells us when two parabolic Mobius maps with distinct fixed points generate a discrete group. This question is settled by the following result of Matelski [12]. As the proof is short, we include it for completeness. ...
The complex hyperbolic version of Shimizu's lemma gives an upper bound on the radii of isometric spheres of maps in a discrete subgroup of PU(n, 1) containing a vertical Heisenberg translation. The purpose of this paper is to show that in a neighbourhood of this bound radii of isometric spheres only take values in a particular discrete set. When the group contains certain ellipto-parabolic maps this upper bound can be improved and the set of values of the radii is more restricted. Examples are given that show that these results cannot be improved.
... Coxeter decompositions of hyperbolic triangles were studied in [10], [11], [12], [13] and [2]. Coxeter decompositions of hyperbolic tetrahedra are listed in [4]. ...
A Coxeter decomposition of a polyhedron in a hyperbolic space ℋn is a decomposition of it into finitely many Coxeter polyhedra such that any two tiles having a common facet are symmetric with respect to it. The classification of Coxeter decompositions is closely related to the problem of the classification of finite-index subgroups generated by reflections in discrete hyperbolic groups generated by reflections. All Coxeter decompositions of simplexes in the hyperbolic spaces ℋn with n > 3 are described in this paper.
... The idea of a trace minimizing algorithm is due to him (see [ 17] and the references given there). Our geometric algorithm builds upon his algebraic algorithm and Matelski's geometric treatment [13]. ...
Let G be a subgroup of PSL(2, C). The discreteness problem for G is the problem of determining whether or not G is discrete. In this paper we assume that all groups considered are non-elementary. If G is generated by two elements, A and B of PSL(2, C), we have what is called the two-generator discreteness problem. Two-generator groups are important because, by a result of Jorgensen, an arbitrary subgroup of PSL(2, C) is discrete if and only if every non-elementary two-generator subgroup is [8]. One solution to the two-generator real discreteness problem (i.e., A and B in PSL(2, R)) is a geometrically motivated algorithm which was begun in [7] and completed in [6], where the algorithm is given in three forms. Our goal here is to compute the computational complexity of the three forms of the PSL(2, R) discreteness algorithm that appear in [6] and of the algorithm restricted to PSL(2, Q). While this may seem far afield from the original mathematical problem of determining discreteness, it settles the question as to in what sense the discreteness problem requires an algorithm. That is, the computational complexity of the algorithm can be used to prove that an algorithmic approach is necessary and that the geometric algorithm of [6] is the best discreteness condition that one can hope to obtain. We also investigate the computational complexity of several other discreteness criteria, including Riley's PSL(2, C) procedure [15], which is not always an algorithm, and Jorgensen's inequality [8]. We find that the geometric two-generator PSL(2, Q) algorithm is of linear complexity. By contrast, Riley's procedure appears to be at least exponential even when restricted to the two-generator PSL(2, Q) case; but it has never been completely analyzed.
... The axis of f has endpoints ±e δ and the axis of g has endpoints ±1. By Reference [11], their axes are disjoint and the distance between the two axes is δ. Moreover, β(f ) = 4 sinh 2 (λ) and β(g) = 4 sinh 2 (µ). ...
This paper is concerned with Loxodromic Mobius Transformations with disjoint
Axes in a Kleinian Group. We study mainly the distance between their axes, and
give some estimates about their translation lengths.
... See figure 2. Denote by R l the reflection in the line l. Proof We use an elegant argument of Matelski in [23]; there is also a proof by computation. Let d g , d h denote the translation distance of g, h. ...
We consider the relationship between hyperbolic cone-manifold structures on
surfaces, and algebraic representations of the fundamental group into a group
of isometries. A hyperbolic cone-manifold structure on a surface, with all
interior cone angles being integer multiples of , determines a holonomy
representation of the fundamental group. We ask, conversely, when a
representation of the fundamental group is the holonomy of a hyperbolic
cone-manifold structure. In this paper we prove results for the punctured
torus; in the sequel, for higher genus surfaces. We show that a representation
of the fundamental group of a punctured torus is a holonomy representation of a
hyperbolic cone-manifold structure with no interior cone points and a single
corner point if and only if it is not virtually abelian. We construct a
pentagonal fundamental domain for hyperbolic structures, from the geometry of a
representation. Our techniques involve the universal covering group of the
group of orientation-preserving isometries of the hyperbolic plane, and Markoff
moves arising from the action of the mapping class group on the character
variety.
... Coxeter decompositions of hyperbolic triangles were studied in [4], [5], [6], [7] and [1]. Coxeter decompositions of hyperbolic simplices of the dimension greater than three were classified in [2]. ...
this paper we classify all Coxeter decompositions of hyperbolic tetrahedra. Definitions
... We observe that Ax [A,B] is disjoint from the axes Ax A and Ax B as follows. Following [14] construct h, the perpendicular from p B to Ax A . By the trace minimality of the stopping generators, h intersects Ax A between p A and A −1 (p) (or vice-versa) [5]. ...
For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a ``hyperelliptic'' three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the {\sl Weierstrass lines} of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six {\sl generalized Weierstrass points} on the surface.
... Since e f , g ⊂ Γ keeps κ 1 invariant and preserves orientation of κ 1 , e f , g acts on κ 1 as a subgroup of PSL(2, R). However, e f , g is not discrete if h = e f g −1 is non-primitive elliptic, which is a rotation through 2kπ/q in our case, by [10] or [11]. Theorem 2.3 is proved. ...
We deal with two-generator subgroups of PSL(2,C) with real traces of both generators and their commutator. We give discreteness criteria for these groups when at least one of the generators is parabolic. We also present a list of the corresponding orbifolds.
... Coxeter decompositions of hyperbolic triangles were studied in [4], [5], [6], [7] and [1]. Coxeter decompositions of hyperbolic simplices of the dimension greater than three were classified in [2]. ...
We classify Coxeter decompositions of hyperbolic tetrahedra, i.e. simplices
in the hyperbolic space H^3. The paper completes the classification of Coxeter
decompositions of hyperbolic simplices.
... Les théories de Markoff deséquations M s1s2 (a, ∂K, u θ ) conduisent très naturellementà définir des générateurs A et B de groupes fuchsiensà deux générateurs. Elles donnent dans le cas parabolique les groupes de Fricke bien connus [681] [528]. On l'a démontré de façon rigoureuse. ...
The text deals with generalizations of the Markoff equation in number theory, arising from continued fractions. It gives the method for the complete resolution of such new equations, and their interpretation in algebra and algebraic geometry. This algebraic approach is completed with an analytical development concerning fuchsian groups. The link with the Teichmuller theory for punctured toruses is completely described, giving their classification with a reduction theory. More general considerations about Riemann surfaces, geodesics and their hamiltonian study are quoted, together with applications in physics, 1/f noise and zeta function. Ideas about important conjectures are presented. Reasons why the Markoff theory appears in different geometrical contexts are given, thanks to decomposition results in the group GL(2,Z).
p>The algebraic theory of maps and hypermaps is summarized in Chapter 1.
There is a group of six invertible topological operations on maps which are induced by automorphisms of a certain Coxeter group that can be identified with an extended Fuchsian triangle group. In Chapter 2 we study how these operations behave with respect to the property of orientability of maps, and we determine the orbits under the group of operations on reflexible torus maps of Euclidean type.
The corresponding groups of operations on hypermaps are infinite, and they partition the sets of symmetrical hypermaps having the same automorphism group into orbits. In Chapter 3, given a group from one of several infinite families (cyclic, dihedral, affine general linear), we use the theory of T -systems to examine how the size of the orbits increase with the size of the group.
In Chapters 4 and 5 we generalize the concept of (hyper)map operations by considering functors induced by more general homomorphisms between triangle groups and extended triangle groups.
The concept of a 2-dimensional algebraic map can be generalized to n dimensions, and the group of operations on n -dimensional maps has order 8 for n > 2. In Chapter 6 we give a combinatorial description of these operations, and examine the orbits of small 3-maps and of certain reflexible n -torus maps. We then consider a further generalization of operations as isomorphism-induced equivalence between categories of different map-like objects. In particular, we exhibit a representation of orientable 3-maps without boundary by (unrestricted) hypermaps, and another of general 3-maps by a certain family of maps.</p
Let N⊂P and P⊂M be inclusions of II 1 factors with finite Jones index. We study the composed inclusion N⊂P⊂M by computing the fusion of N-P and P-M bimodules and determine various properties of N⊂M in terms of the “small” inclusions. A nice class of such subfactors arises in the following way: let H and K be two finite groups acting properly outerly on the hyperfinite II 1 factor M and consider the inclusion M H ⊂M⋊K. We show that properties like irreducibility, finite depth, amenability and strong amenability (in the sense of Popa) of M H ⊂M⋊K can be expressed in terms of properties of the group G generated by H and K in OutM. In particular, the inclusion is amenable iff M is hyperfinite and the group G is amenable. We obtain many new examples of infinite depth subfactors (amenable and nonamenable ones), whose principal graphs have subexponential and/or exponential growth and can be determined explicitly. Furthermore, we construct irreducible, amenable subfactors of the hyperfinite II 1 factor which are not strongly amenable.
We embed the Teichmüller space of the once punctured torus T
(1,1) into the set of conjugacy classes of groups generated by three anti-holomorphic involutions I
1, I
2 and I
3 (Lagrangian triangle groups), acting on the complex hyperbolic plane . We deform this embedding, and obtain a three dimensional family E of discrete, faithful and type preserving representations of the fundamental group of the once punctured torus.
In this paper we give necessary and sufficient conditions for discrete- ness of a subgroup of PSL(2, C) generated by a hyperbolic element and an elliptic one of odd order with non-orthogonally intersecting axes. Thus we completely determine two-generator non-elementary Kleinian groups without invariant plane with real traces of the generators and their com- mutator. We also give a list of all parameters that correspond to such groups. An interesting corollary of the result is that the group of the minimal known volume hyperbolic orbifold H3/ 353 has real parameters.
We give a complete condition for any n elements of PSL(2, R) to generate a Fuchsian group which is a Schottky group on that set of generators, and apply it to a
question of Bowditch on representations of surface groups into PSL(2, R). 2000 Mathematical Subject Classification: 20H10, 32G15.
Let f and g be elements of the isometry group Isom(H2) of the hyperbolic plane H2, and assume that one of them is orientation-reversing. We determine when the group (f,g) they generate is discrete; in particular, we obtain the classification of such groups. As an application to knot theory, we completely determine the tunnel number one Montesinos knots.
In this paper a new discreteness criterion for subgroups in SL(2,
C) is established,
which implies all known discreteness criteria by two-generator subgroup
and is the
best possible. And it is proved that every non-elementary and non-discrete
subgroup
of SL(2, C) has a non-elementary and non-discrete subgroup generated by
two
loxodromic elements.
Recently Gehring, Gilman, and Martin introduced an important class of two-generator groups with real parameters:
{ G =
f,g
| f,g Î PSL(2,C);b,b¢,g Î R }\left\{ {\Gamma = \left\langle {f,g} \right\rangle \left| {f,g \in PSL(2,C);\beta ,\beta ^\prime ,\gamma \in R} \right.} \right\}
whereβ=tr2
f−4,β′=tr2
g−4, and γ=tr(fgf
−1
g
−1)−2. The groups that belong to this class we callRP groups. We find criteria for discreteness ofRP groups generated by a hyperbolic element and an elliptic one of even order with intersecting axes.
We consider three types of Schottky spaces which consist of non- Fuchsian classical Schottky groups of real type of genus two. This paper has the following two aims: (1) to represent the shape of the spaces by using multipliers and cross ratios of the fixed points of two generators of marked Schottky groups; (2) to determine fundamental regions for the Schottky modular group of genus two acting on the spaces.
Recently Gehring, Gilman, and Martin introduced an important class of two-generator groups with real parameters: f = hf; gi j f; g 2 PSL(2; C); ; 0 ; 2 Rg; where = tr 2 f 4, 0 = tr 2 g 4, and = tr(fgf 1 g 1 ) 2. The groups that belong to this class we call RP groups. We nd criteria for discreteness of RP groups generated by a hyperbolic element and an elliptic one of even order with intersecting axes.
. In this article we continue investigation of a special class of 2generator Kleinian groups in which neither of the generators nor their commutator is strictly loxodromic. We give necessary and sufficient conditions for discreteness of non-elementary and non-Fuchsian groups of this class generated by two hyperbolic elements. Introduction Let Gamma = hA; Bi be a two-generator subgroup of PSL(2; C ). We study the following question: when Gamma is discrete? The same question for two-generator subgroups of PSL(2; R) has a problematic history in the mathematical literature. There were a lot of papers dealing with this problem, see [Br2], [DJ], [Gi1] -- [Gi3], [GiM], [J], [KIR], [Kn], [LU], [Mt], [P1] -- [P3], [PR], [R1] -- [R5]. Rosenberger [R5] completed the classification of all two-generator Fuchsian groups and presented an algebraic algorithm for determining whether or not a two-generator subgroup of PSL(2; R) is discrete. Another, geometric approach to this problem was started by M...
Let A, BsLf(2,1R), the group of linear fractional transformations with real entries and determinant 1. The results of Knapp [2], Purzitsky [8], and Rosenberger [10] give conditions so that we may determine in all cases but one whether the group {A, B} generated by A and B is a Fuchsian group. The remaining case is when A and B are hyperbolic transformations whose commutator [A, B] is of finite order. We give necessary and sufficient conditions for this group to be discrete. We also obtain all faithful representations of a group whose presentation is {A, B] [A, B]" = 1 } by a discrete subgroup of PSL(2, IR) = SL(2, IR)/{ ___ 1 } ~Lf(2,1R) and partition the representations into disjoint conjugacy classes. Here 1 is the identity. Results in this direction were first given by Lehner and Newman [4, 5] who considered the case of two elliptic generators. Their results were extended to all two generator free products by Purzitsky [-9] and Rosenberger [10]. The form of these distinct conjugacy classes provides an explicit solution to a problem which appears in [1]. Let