No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
All generating pairs of all two-generator Fuchsian groups
... For /?, m ^ 2 Satz 2.2. of [8] gives a complete (negative) answer to the first (and hence to the second) question (see the above final remark). On the other hand there are exotic generating systems known for p = 0 and m = 3 (see [9] ) and others conjectured for m ^ 4, p = 0, 1. In order to narrow down the margin for possible non-standard phenomena we would at least like to state the following Remark; its proof is a somewhat complicated extension of the proof of the above mentioned Satz 2.2. of [8] (see [10] ). ...
... In order to narrow down the margin for possible non-standard phenomena we would at least like to state the following Remark; its proof is a somewhat complicated extension of the proof of the above mentioned Satz 2.2. of [8] (see [10] ). (For p = 0 and m^6a proof can be found also in [5], and for p = 0 and m = 3 it follows from [9]. For p = 1 the above Remark can be strengthened further to cases where one requires only 7r(i) i? 3 for at least two of the i.) ...
... About question (2) above we would like to point out that for the case p = 0, m = 3 and pairwise distinct ir(i) the group G is always rank 2 fully almost quasi free (defined in the above final remark), although for low 77(0 there exist frequently exotic Nielsen classes of generating pairs: These are listed in [9], Theorem 1 and the succeeding Remark 2. The above claim follows from the classification given there and the known facts about Out G for rank G = 2. ...
This paper has been motivated by earlier work of the first two authors (see [3] ), where distinct Nielsen classes of generating systems for a Fuchsian group have been established and, in the case of odd and pairwise relative prime exponents π(i), classified. As a consequence they could distinguish nonisotopic Heegaard decompositions of Seifert fibred 3-manifolds. In proving that these decompositions are actually non-homeomorphic (see [3], Section 2), they investigated the question whether the different Nielsen classes of generating systems for G remain distinct, if one passes over to the weaker notion of “Nielsen equivalence up to automorphisms” (see [12], p. 3.5, 4.11 a-c): this means that the automorphisms of G are added to the Nielsen equivalence relations on the generators.
... It is an interesting problem to study when a subgroup generated by two non-commuting parabolic elements is free. A similar problem in the case of Fuchsian and Kleinian groups has been studied by many authors [4,6,7,9,10]. They have explored the conditions for two elements in Fuchsian or Kleinian groups to generate a discrete free group. ...
... They have explored the conditions for two elements in Fuchsian or Kleinian groups to generate a discrete free group. We suggest interested readers to consult with many works of Fine and Rosenberger on this topic, especially surveys [4,10] for more references on this topic. A particular case of this classical problem is to find conditions when a subgroup G of PSL(2, C) generated by two non-commuting, parabolic, linear fractional transformations ...
Let A and B be two Heisenberg translations of PU(2,1). In this paper, we will discuss the groups ⟨A,B⟩ generated by two non-commuting Heisenberg translations and determine when they are free. The main result of the paper improves an assertion made by Xie et al. (Canad. Math. Bull.56(4) (2013) 881–889). We also extend the improved result for two Heisenberg translations in PSp(2,1).
... We say that two n-tuples T and T are Nielsen equivalent if there exists a finite sequence of n-tuples T = T 0 , T 1 , . . . , T k−1 , T k = T such that T i is obtained from T i−1 by an elementary Nielsen transformation for 1 i k. Nielsen equivalence clearly defines an equivalence relation for tuples of elements of G. Nielsen equivalence in Fuchsian groups have been extensively studied by many authors, see [2,5,6,7,8,11,12,13,15] for example. However the techniques deployed so far are mainly algebraic such as normal form and K-theoretic arguments. ...
... 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]. ...
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 ...
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.
... Proof. The necessity of (1) follows from67 . For the proof of the sufficiency , by [4], we may assume that E(G, G)\neq\emptyset , i.e., G contains some elliptic elements of order at least 3, cf. ...
... Proof. By [6] , the necessity of (1) is obvious. For the sufficiency, we suppose that \inf\{|A|:A\in TH(G, G)\}\geq c_{0} , but G is not discrete. ...
... 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]. ...
... The axes of hyperbolic generators project onto closed geodesics on the quotient manifold; parabolic elements appear if there are punctures. Discreteness questions in the intersecting axes cases are treated in [2,8,9,16]. ...
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.
... 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.
... Question of root existence in different forms appears in the Purtzitsky-Rosenberger trace minimizing algorithm, [4], [19], it was considered roots and rational powers of one or both generators of non-elementary two generator discrete subgroups of P SL 2 (R) found by the GM algorithm. But we solve existing root problem for arbitrary element of SL 2 (F p ). ...
The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.
... Question of root existing in different forms appears in the Purtzitsky-Rosenberger trace minimizing algorithm [33,35] it was considered roots and rational powers of one or both generators of in non-elementary two generator discrete subgroups of P SL 2 (R) found by the GM algorithm. But we solve existing root problem for arbitrary element of P SL 2 (F p ). ...
We introduce Extended special linear group , which is generalization of matrix group over perfect field k. Extended special linear group , where k is arbitrary perfect field, is storage of all square matrix roots from . The analytical formula of square roots of 2-nd and 4-th power in , , for any prime p, as well as in , and in , where k is arbitrary perfect field, is found by us. New linear group which is storage of square roots from is found and investigated by us. The criterion of roots existing for different classes of matrix -- simple and semisimple matrixes from , are established. The problems of square root from group element existing in , and for arbitrary prime p are solved in this paper. The similar goal of root finding was reached in the GM algorithm adjoining an n-th root of a generator \cite{GM} results in a discrete group for group SL(2,R), but we consider this question over finite field . Over method gives answer about existing without exponenting M to n-th power. We only use the trace of M or only eigenvalues of M. In \cite{Amit} only the Anisotropic case of group , where Q is a quaternion division algebra over k was considered. The authors of \cite{Amit} considered criterion to be square only for the case is a field of characteristics not equal 2. We solve this problem even for fields and . The criterion to be square in was not found by them what was declared in a separate sentence in \cite{Amit}. We consider more general case \cite{SkSq} consisting in whole group .
... The analysis of ergodicity is more complicated. The process must be periodic if the subgroup of SL(2, R) generated by the steps is discrete and two-generator discrete subgroups of SL(2, R) have been classified [155,156]. This is related to Furstenberg's theorem [157] and was discussed in this context in [90]. ...
Driven quantum systems exhibit a large variety of interesting and sometimes exotic phenomena. Of particular interest are driven conformal field theories (CFTs) which describe quantum many-body systems at criticality. In this paper, we develop both a spacetime and a quantum information geometry perspective on driven 2d CFTs. We show that for a large class of driving protocols the theories admit an alternative but equivalent formulation in terms of a CFT defined on a spacetime with a time-dependent metric. We prove this equivalence both in the operator formulation as well as in the path integral description of the theory. A complementary quantum information geometric perspective for driven 2d CFTs employs the so-called Bogoliubov-Kubo-Mori (BKM) metric, which is the counterpart of the Fisher metric of classical information theory, and which is obtained from a perturbative expansion of relative entropy. We compute the BKM metric for the universal sector of Virasoro excitations of a thermal state, which captures a large class of driving protocols, and find it to be a useful tool to classify and characterize different types of driving. For M\"obius driving by the SL(2,R) subgroup, the BKM metric becomes the hyperbolic metric on the unit disk. We show how the non-trivial dynamics of Floquet driven CFTs is encoded in the BKM geometry via M\"obius transformations. This allows us to identify ergodic and non-ergodic regimes in the driving. We also explain how holographic driven CFTs are dual to driven BTZ black holes with evolving horizons. The deformation of the black hole horizon towards and away from the asymptotic boundary provides a holographic understanding of heating and cooling in Floquet CFTs.
... 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 show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking.
... The problem of identifying discrete two-generator subgroups of PSL 2 (R) has been extensively studied in the literature. A complete classification of such groups, and an algorithm to decide whether or not a two-generator subgroup of PSL 2 (R) is discrete, is given in [10,18]. ...
Let K be a non-archimedean local field with residue field of characteristic p. We give necessary and sufficient conditions for a two-generator subgroup G of to be discrete, where either or G contains no elements of order p. We also present a practical algorithm which decides whether such a subgroup G is discrete. A crucial ingredient for this work is a general structure theorem for two-generator groups acting by isometries on a -tree, obtained by applying Klein-Maskit combination theorems.
... The case of two generator subgroups of PSL(2, R) has been completely resolved by the work of Gilman [56,58]. Once discrete, all isomorphism types of two-generator Fuchsian groups can be found in earlier work of Purzitsky and Rosenberger, [121]. ...
The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in , and has a natural coordinate system (introduced by Linda Keen and Caroline Series in the early 1990s) which reflects the geometry of the underlying 3-manifold deformations. The Riley slice arises in the study of arithmetic Kleinian groups, the theory of two-bridge knots, the theory of Schottky groups, and the theory of hyperbolic 3-manifolds; because of its simplicity it provides an easy source of examples and deep questions related to these subjects. We give an introduction for the non-expert to the Riley slice and much of the related background material, assuming only graduate level complex analysis and topology; we review the history of and literature surrounding the Riley slice; and we announce some results of our own, extending the work of Keen and Series to the one complex dimensional moduli spaces of Kleinian groups isomorphic to acting on the Riemann sphere, . The Riley slice is the case (i.e. two parabolic generators).
... The question has been raised as to whether this translates to an algebraic treatment using the Purtzitsky-Rosenberger trace minimizing algorithm [12][13][14]. The trace minimizing method is to replace (A, L B 1/n ) when T r A ≥ T r L B 1/n by one of the ordered pairs (L B 1/n , AL B 1/n ), (L B 1/n , AL −1 B 1/n ), (AL B 1/n , L B 1/n ) or (AL −1 B 1/n , L B 1/n ) depending upon the sizes of the traces. ...
Let G be a finitely generated group of isometries of , hyperbolic 2-space. The discreteness problem is to determine whether or not G is discrete. Even in the case of a two generator non-elementary subgroup of (equivalently ) the problem requires an algorithm (Gilman and Maskit in Mich Math J 38:13–32, 1991; Gilman in two-generator discrete subgroups of , 1995). If G is discrete, one can ask when adjoining an nth root of a generator results in a discrete group. In this paper we first address the issue for pairs of hyperbolic generators in with disjoint axes and obtain necessary and sufficient conditions for adjoining roots for the case when the two hyperbolics have a hyperbolic product and are what as known as stopping generators for the Gilman–Maskit algorithm (Gilman and Maskit 1991; Gilman and Keen in Cutting sequences and palindromes in geometry of Riemann surfaces, 2009). Stopping generators are generators that correspond to certain generalized Coxeter triangle group. We give an algorithmic solution in the other cases that are not stopping generators. Our solution applies to all other types of pairs of stopping generators that arise in what is known as the intertwining case. The results are geometrically motivated and stated as such, but also can be given computationally using the corresponding matrices.
... For an elaborate account on this problem, see Gilman [11]. Algorithmic solutions to this problem were given by Rosenberger [20], Gilman and Maskit [12], Gilman [11]. The Jørgensen inequality [7] is a major result related to this problem. ...
Let SL(2;IH) be the group of 2 × 2 quaternionic matrices with Dieudonné determinant one. The group SL(2;IH) acts on the five dimensional hyperbolic space by isometries. We investigate extremality of Jørgensen type inequalities in SL(2;IH). Along the way, we derive Jørgensen type inequalities for quaternionic Möbius transformations which extend earlier inequalities obtained by Waterman and Kellerhals.
... A solution even in the P SL(2, R) case appears to require an algorithm that relies on a the concept of trace minimizing. This concept was initiated and its use pioneered by Rosenberger and Purzitsky in the 1970's [17,18,20,19] Their work has lead to many discreteness results and algorithms. For a more recent summary see [7]. ...
... Gilman and Maskit, [6,14], building upon work of [25,26,27] solved the discreteness problem for two generator subgroups of Isom(H 2 ) by producing Date: August 5, 2015. Some of this work was carried out while the author was a visitor at ICERM. ...
The discreteness problem, that is, the problem of determining whether or not
a given finitely generated group G of orientation preserving isometries of
hyperbolic three-space is discrete as a subgroup of the whole isometry group of
hyperbolic three space, is a challenging problem that has been investigated for
more than a century and is still open. It is known that G is discrete if, and
only if, every non-elementary two generator subgroup is. Several sufficient
conditions for discreteness are also known as are some necessary conditions,
though no single necessary and sufficient condition is known. There is a finite
discreteness algorithm for the two generator subgroups of the isometry group of
hyperbolic two-space. But the situation in three dimensions is more delicate
because there are geometrically infinite groups.
We present a semi-algorithm, that is, a procedure that terminates sometimes
but not always. There is no standard way to find an infinite sequence of
distinct elements that converges to the identity to show that a group is not
discrete. Our semi-algorithm either produces such an infinite sequence or finds
a finite sequence that produces a right angled hexagon in hyperbolic
three-space which has a special property that is a generalization of the notion
of convexity. We call it a canonical hexagon. If the group is discrete, free
and geometrically finite, it always has an essentially unique canonical hexagon
which the procedure finds in a finite number of steps.
... The first step of the algorithm determines whether the axes of the generators intersect or not. This is determined by computing the trace of the commutator, [C, D] = CDC −1 D −1 (see [4] or [17]). If T r [C, D] < −2, the axes intersect; if −2 < T r [C, D] < 2, the group contains an elliptic element; and if T r [C, D] > 2, the axes are disjoint. ...
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.
... For an elaborate account of this problem, see Gilman [8]. Algorithmic solutions to this problem were given by Rosenberger [17], Gilman and Maskit [9], Gilman [8]. The Jørgensen inequality [5] is one of the major results related to this problem. ...
Let be the group of quaternionic
matrices with Dieudonn\'e determinant 1. The group
acts on the five dimensional hyperbolic space by isometries. We derive
necessary conditions for discreteness of two-generator subgroups of . These are J{\o}rgensen type inequalities for quaternionic
M\"obius transformations and extensions of results by Waterman and Kellerhals.
We further investigate extremality of J{\o}rgensen type inequalities in .
The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in ℂ, and has a natural coordinate system (introduced by Linda Keen and Caroline Series in the early 1990s) which reflects the geometry of the underlying 3-manifold deformations. The Riley slice arises in the study of arithmetic Kleinian groups, the theory of two-bridge knots, the theory of Schottky groups, and the theory of hyperbolic 3-manifolds; because of its simplicity it provides an easy source of examples and deep questions related to these subjects. We give an introduction for the non-expert to the Riley slice and much of the related background material, assuming only graduate level complex analysis and topology; we review the history of and literature surrounding the Riley slice; and we announce some results of our own, extending the work of Keen and Series to the one complex dimensional moduli spaces of Kleinian groups isomorphic to ℤ/aℤ*ℤ/bℤ acting on the Riemann sphere, 2 ≤ a,b ≤ ∞. The Riley slice is the case of two parabolic generators where a = b = ∞.
A bstract
Driven quantum systems exhibit a large variety of interesting and sometimes exotic phenomena. Of particular interest are driven conformal field theories (CFTs) which describe quantum many-body systems at criticality. In this paper, we develop both a spacetime and a quantum information geometry perspective on driven 2d CFTs. We show that for a large class of driving protocols the theories admit an alternative but equivalent formulation in terms of a CFT defined on a spacetime with a time-dependent metric. We prove this equivalence both in the operator formulation as well as in the path integral description of the theory. A complementary quantum information geometric perspective for driven 2d CFTs employs the so-called Bogoliubov-Kubo-Mori (BKM) metric, which is the counterpart of the Fisher metric of classical information theory, and which is obtained from a perturbative expansion of relative entropy. We compute the BKM metric for the universal sector of Virasoro excitations of a thermal state, which captures a large class of driving protocols, and find it to be a useful tool to classify and characterize different types of driving. For Möbius driving by the SL(2 , ℝ) subgroup, the BKM metric becomes the hyperbolic metric on the disk. We show how the non-trivial dynamics of Floquet driven CFTs is encoded in the BKM geometry via Möbius transformations. This allows us to identify ergodic and non-ergodic regimes in the driving. We also explain how holographic driven CFTs are dual to driven BTZ black holes with evolving horizons. The deformation of the black hole horizon towards and away from the asymptotic boundary provides a holographic understanding of heating and cooling in Floquet CFTs.
L. Louder showed that any generating tuple of a surface group is Nielsen equivalent to a stabilized standard generating tuple i.e. where is the standard generating tuple. This implies in particular that irreducible generating tuples, i.e. tuples that are not Nielsen equivalent to a tuple of the form , are minimal. In a previous work the first author generalized Louder's ideas and showed that all irreducible and non-standard generating tuples of sufficiently large Fuchsian groups can be represented by so-called almost orbifold covers endowed with a rigid generating tuple. In the present paper a variation of the ideas from \cite{W2} is used to show that this almost orbifold cover with a rigid generating tuple is unique up to the appropriate equivalence. It is moreover shown that any such generating tuple is irreducible. This provides a way to exhibit many Nielsen classes of non-minimal irreducible generating tuples for Fuchsian groups. As an application we show that generating tuples of fundamental groups of Haken Seifert manifolds corresponding to irreducible horizontal Heegaard splittings are irreducible.
We provide a direct connection between the (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank 2. We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the -JSJ decomposition of such groups.
We describe a practical algorithm to solve the constructive membership problem for discrete two-generator subgroups of or . This algorithm has been implemented in Magma for groups defined over real algebraic number fields.
It is proven by using Lyapunov exponents that the Teichmüller curves from Chap. 8 are never twists of Teichmüller curves in M2.
We describe the relation between the stabilizer of the graph of the Teichmüller curve and the commensurator of the Veech group. Furthermore we introduce the notion of pseudo parabolic maximal Fuchsian groups and show why this property is useful for calculating the stabilizer.
We give an overview of the basic concepts which are used everywhere else in these notes. We recall well-known results about real quadratic number fields, Fuchsian groups, moduli spaces and Hilbert modular surfaces. This chapter will be helpful in particular for people who are not already familiar with the mentioned topics.
We recall McMullen’s construction of Teichmüller curves in M3 and M4 using Prym varieties. Also these Teichmüller curves yield Kobayashi curves on XD.
The volume of twisted Teichmüller curves is calculated for most matrices M if the class number of OD is equal to 1. From this the classification of twisted Teichmüller curves can be derived. Finally, we present some ideas how quantities like the number of elliptic fixed points, the number of cusps and the genus of twisted Teichmüller curves can be calculated in some special cases.
In this chapter we introduce Teichmüller curves. In particular, we recall Calta’s and McMullen’s construction of Teichmüller curves in M2, which was already mentioned above.
We recall Oseledet’s Theorem on the existence of Lyapunov exponents and introduce the Kontsevich-Zorich cocycle over Teichmüller curves. Furthermore the connection of the Teichmüller flow to the geodesic flow on T1H is discussed.
The main new objects of these notes, namely twisted Teichmüller curves, are defined. Only some main properties of twisted Teichmüller curves are derived here. Most importantly, it is shown that twisted Teichmüller curves yield indeed Kobayashi curves.
When asked recently about how he does mathematics, Paul Halmos replied “Since a lot of my work has been in operator theory and infinite dimensional Hilbert spaces, and since the most easily accessible part of that is matrix theory and finite dimensional vector spaces, I start by looking at a 2 by 2 matrix.”
This paper is intended as a survey of the recent joint work of F.W. Gehring with the author and is an expanded version of the lecture given at the conference “Quasiconformal Mappings and Analysis.”
Let F bea non-elementary subgroup of PSL(2, ℝ). The purpose of this paper is to elucidate the nature and the extent of elliptic elements in F.
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 study the class of two-generator subgroups of PSL(2, C) with real parameters which was introduced recently by Gehring, Gilman, and Martin. We give criteria for discreteness of non-elementary and non-Fuchsian groups of this class that are generated by two hyperbolic elements. We construct all hyperbolic orbifolds uniformized by the discrete groups of such type. The orbifolds described are of infinite volume.
An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G. We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group A5A5, and give various other examples.
We present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.
M. Newman [8] gave a complete group-theoretic description of the normal subgroups of genus one of the modular group PSL(2,Z) ≅ Z2*Z3. In this paper we generalize his result and give a characterization of the normal subgroups of genus one of free products of finitely many finite cyclic groups. In particular we give a complete group-theoretic description of the normal subgroups of genus one of the next important Hecke-group (Formula presented.). As an interesting corollary we get the following. If b_(n) is the number of normal subgroups of (Formula presented.) of index kn and genus 1 (n ∈ N) then (Formula presented.)
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.
Nielsen equivalence and simple-homotopy equivalence are interpreted as analogues in dimensions 1 and 2 respectively. This yields the following results. § 1. A new method for distinguishing Nielsen equivalence classes of generating systems in a group G is presented. As an application, inequivalent generating systems are exhibited in metabelian groups and in cocompact Fuchsian groups (extending the results of [12]). § 2. The techniques developed in § 1 are applied to the simple-homotopy theory of 2-complexes, and examples of pairs of 2-complexes K, L are constructed, which are homotopy equivalent, but not simple-homotopy equivalent. The simplest consists of the one-vertex 2-complexes for the presentations K: (x, y, z y3, yx10 y-1 x-5, [x7, z]), L: (x, y, z y3, yx10y-1x-5, x14zx14z-1x-7zx-21z-1).
Let Γ be a group of Möbius transformations of the unit disk D or of the upper halfplane onto itself. The results of C. L. Siegel [15] and of A. Marden [10] show that, leaving aside the elementary groups, there is a sharp dividing line between discrete (Fuchsian) and non-discrete groups. The aim of this paper is to give a sharp quantitative form of Marden’s theorem (see Section 4).
This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known (but virtually unobtainable) Fenchel-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo metrical ideas to be found in that manuscript, as well as some more recent material. The text has been written with the conviction that geometrical explana tions are essential for a full understanding of the material and that however simple a matrix proof might seem, a geometric proof is almost certainly more profitable. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry. However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical (not axiomatic) hyperbolic geometry. It is hoped that this chapter will serve as a "dictionary" offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right.
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.
Besitzt eine endlich erzeugte, diskrete Gruppe G von Automorphismen der oberen Halbebene einen kompakten Fundamentalbereich, so läßt sich G darstellen durch
mit \mu (G) = 2g - 2 + \mathop \sum \limits_{i = 1}^r (1 - \frac{1}{{{n_1}}}) > 0,{n_i} \geq 2,g \geq 0,r \geq 0, wobei .
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
The geometry of discrete groups. Graduate Texts in Math
- A F Beardon