Bernard Maskit

On Klein’s combination theorem. IV

This paper contains an expansion of the combination theorems to cover the following problems. New rank 1 parabolic subgroups are produced, while, as in previous versions, all elliptic and parabolic elements are tracked. A proof is given that the combined group is analytically finite if and only if the original groups are; in the analytically finite case, we also give a formula for the hyperbolic area of the combined group (i.e., the hyperbolic area of the set of discontinuity on the 2-sphere modulo G) in terms of the hyperbolic areas of the original groups. There is also a new variation on the first combination theorem in which the common subgroup has finite index in one of the two groups

Beneath the Surface of the Therapeutic Interaction: the Psychoanalytic Method in Modern Dress

This study represents a new generation of psychotherapy process research, using multiple perspectives on the data of the analytic situation, including impressions of the treating analyst, ratings of complete sessions by clinical judges, and objective linguistic measures. Computerized measures of language style developed in the framework of multiple code theory were applied to verbatim session recordings from a psychoanalytic case; the measures are illustrated in microanalyses of the process in two sessions. The results show agreement between the linguistic measures and clinical ratings based on a psychoanalytic perspective. The linguistic measures look beneath the surface of the therapeutic interaction by relying largely on lexical items of which clinicians are not likely to be explicitly aware, and enable a new perspective on the therapeutic discourse as seen in the graphic images of the microprocess. While the results of this study were limited to a single case, the automatized measures can be readily applied to large samples and in repeated single case designs. Two goals of process research, using measures such as those developed in this study, are discussed: to develop measures of mediating variables that can be used to identify specific treatment effects in comparative outcome studies; and, beyond this pragmatic aim, to assess development of capacities for self-exploration and self-regulation as psychoanalytic treatment goals.

Parabolic elements in Kleinian groups

Let G be a Kleinian group, and T(G) its deformation space. We are concerned here with the question of which loxodromic (including hyperbolic) elements of G can be made parabolic on the boundary of T(G). For some geometrically finite boundary groups (see [8]), one obtains necessary conditions in terms of primitive elements of G represented by simple disjoint loops on U(G)/G. In this paper we restrict our attention to geometrically finite function groups (including groups with torsion), and show that these necessary conditions are sufficient. The restriction to function groups is to some extent a matter of convenience; our techniques can be applied in certain more general situations. The groups we obtain as boundary groups are all geometrically finite. Nothing is known about which elements can be made parabolic at boundary groups which are not geometrically finite. If G is Fuchsian acting on the upper half plane U, then every element of G representing a simple loop on U/G is primitive. Abikoff [1] and Marden, in the torsion-free case (unpublished), showed that a set of elements representing simple disjoint loops on U/G can be made parabolic on the boundary of the deformation space of G, where the deformations are all supported on U. Similar results along the lines of this paper have also been obtained by Thurston (unpublished). The author wishes to thank T. Jorgensen, L. Keen and I. Kra for informative conversations.

Moduli of marked Riemann surfaces

The purpose of this note is to exhibit a set of complex analytic moduli for the space of closed Riemann surfaces of genus g§^2, marked by a basis for the fundamental group. That one could find such moduli (i.e., biholomorphically embed the Teichmüller space of genus g in C*~) was proven by Bers [2]. His moduli are variational—they depend on a choice of base surface. Our moduli are in some sense intrinsic, similar to the (real) moduli of Fenchel-Nielsen [4] and Keen [5]. In fact our moduli should be regarded as the complex analogue of the Fenchel-Nielsen and Keen moduli. (The geometric relationship between these different moduli is clear, and they are real-analytically equivalent.) The actual expressions for the moduli given below involve multiplicative constants and square roots. These normalizations serve two purposes. First, the moduli space is contained in a product of half-planes. Second, with these normalizations, the group of translations

Construction of Kleinian Groups

The purpose of this paper is to investigate a special case of the following Conjecture. Let D be a plane domain, and let G he a group of conformai autohomeomorphisms of D. Then there exists a schlicht function f, mapping D onto a plane domain D’, so that every element of f o G o f -1 is a Mobius transformation.

A Weighted Referential Activity Dictionary

The Weighted Referential Activity Dictionary (WRAD) is a dictionary (word list) containing 696 items, with weights ranging between −1 and +1, used for computer modeling of a psycholinguistic variable, Referential Activity (RA), in spoken and written language. The RA dimension concerns the degree to which language reflects connection to nonverbal experience, including imagery, and bodily and emotional experience, and evokes corresponding experience in the listener or reader. RA is primarily indicated by attributes of language style independent of content. High RA language is vivid and evocative; low RA language may be abstract, general, vague or diffuse. RA ratings have been widely used in psycholinguistic and clinical research. RA was initially measured using scales scored by judges; the CRA (Mergenthaler and Bucci, 1999), a binary dictionary, was the first computerized RA measure developed to model judges’ RA ratings. The WRAD, a weighted dictionary, shows higher correlations with RA ratings in all text types tested. The development of the WRAD and its applications are made possible by the authors’ Discourse Attributes Analysis Program (DAAP), which uses smooth local weighted averaging to capture the ebb and flow of RA and similar variables.

Linguistic measures of the referential process in psychodynamic treatment: The English and Italian versions

Abstract The referential process is defined in the context of Buccis multiple code theory as the process by which nonverbal experience is connected to language. The English computerized measures of the referential process, which have been applied in psychotherapy research, include the Weighted Referential Activity Dictionary (WRAD), and measures of Reflection, Affect and Disfluency. This paper presents the development of the Italian version of the IWRAD by modeling Italian texts scored by judges, and shows the application of the IWRAD and other Italian measures in three psychodynamic treatments evaluated for personality change using the Shedler-Westen Assessment Procedure (SWAP-200). Clinical predictions based on applications of the English measures were supported.

New parameters for Fuchsian groups of genus 2

We give a new real-analytic embedding of the Teichmiiller space of closed Riemann surfaces of genus 2 into R6. The parameters are explicitly defined in terms of the underlying hyperbolic geometry. The embedding is accomplished by writing down four matrices in PSL(2, ), where the entries in these matrices are explicit algebraic functions of the parameters. Explicit inequalities are given to define the image of the embedding; the four matrices corresponding to a point in this image generate a fuchsian group representing a closed Riemann surface of genus 2. In this note we introduce a new, canonical, real-analytic embedding of the Teichmiiller space T2, of closed Riemann surfaces of genus 2, onto an explicitly defined region R C R6. The embedding is defined in terms of the underlying hyperbolic geometry; in particular, the parameters are elementary functions of lengths of simple closed geodesics, and angles and distances between simple closed geodesics. We start with a specific marked Riemann surface So, and a specific set of normalized (non-standard) generators, ao, bo, co, do E PSL(2, R), for the fuchsian group Go representing So. Then we can realize a point in T2 as a set of appropriately normalized generators a, b, c, d E PSL(2, I) for the fuchsian group G representing a deformation S of So. We write the entries in the generators, a,... , d, as elementary functions of eight parameters, all defined in terms of the underlying hyperbolic geometry, and we write down explicit formulae expressing two of these parameters as functions of the other six. Three of our six parameters are necessarily positive; we give two additional inequalities to obtain necessary and sufficient conditions for the group G C PSL(2, R), generated by a,... ,d, to be an appropriately normalized quasiconformal deformation of Go. There is a related embedding in [3], where the parameters are fixed points of hyperbolic elements of G. As in [3], we identify the Teichmiiller space with DY), the identity component of the space of discrete faithful representations of 7rl(So) into PSL(2, ) modulo conjugation. The main differences between these two embeddings is that in [3] the four matrices do not have unit determinant and the parameters there are defined in terms of fixed points of elements of the group; here our matrices do have unit determinant, and the parameters are defined in terms of the underlying hyperbolic geometry. Received by the editors October 20, 1997 and, in revised form, February 20, 1998. 1991 Mathematics Subject Classification. Primary 30F10; Secondary 32G15. Research supported in part by NSF Grant DMS 9500557. (?1999 American Mathematical Society

Parameters for Fuchsian Groups I: Signature (0, 4)

This is the first of a series of notes presenting new parameters for certain torsion-free finitely generated Fuchsian and quasifuchsian groups. In this note we consider signature (0, 4). Other low signatures, as well as the general case, will be dealt with elsewhere. Every Fuchsian group of signature (0, 4), acting on the upper half-plane ∪, can be generated by four parabolic transformations, A,B,C,D, where the product ABCD = 1. Normalize so that AB has its attracting fixed point at ∞, its repelling fixed point at 0, and so that the fixed point of C is at 1. Let x be the fixed point of D, and let y be the fixed point of B. Then x > 1 and y < 0. We show that x and y serve as parameters for the deformation space of these groups (this is really two results, one having to do with Fuchsian, and the other with quasifuchsian groups). We also explicitly write the matrices A,B,C,D in PGL(2,ℝ) + (these are 2 × 2 real matrices with positive determinant) as functions of x and y; this gives an explicit example of a stratification (see [K-M]). We also construct an explicit fundamental domain for the Teichmuller modular group for signature (0, 4), and we identify the side pairing transformations.

Space form isometries as commutators and products of involutions

In dimensions 2 and 3 it is well known that given two orientationpreserving hyperbolic isometries that generate a non-elementary group, one can find a triple of involutions so that each isometry can be expressed as a product of two of the three involutions; in this case, we say that the isometries are linked. In this paper, we investigate the extent to which a pair of isometries in higher dimensions can be linked. This question separates naturally into two parts. In the first part, we determine the least number of involutions needed to express an isometry as a product, and give two applications of our results; the second part is devoted to the question of linking. In general, the commutator (involution) length of a group element is the least number of elements needed to express that element as a product of commutators (involutions), and the commutator (involution) length of the group is the supremum over all commutator (involution) lengths. Let Gn be the group of orientation-preserving isometries of one of the space forms, the (n − 1)sphere, Euclidean n-space, hyperbolic n-space. For n ≥ 3, we show that the commutator length of Gn is 1; i.e., every element of Gn is a commutator. We also show that every element of Gn can be written as a product of two involutions, not necessarily orientation-preserving; and, depending on the particular space and on the congruence class of n mod 4, the involution length of Gn is either 2 or 3. In the second part of the paper, we show that all pairs in SO4 are linked but that the generic pair in the orientation-preserving isometries of hyperbolic 4-space or in Gn, n ≥ 5, is not.