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Topic: Kleinian group

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Discrete group

Fuchsian group

SL(2,Z)

Projective linear group

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	Edward Taylor's Publications (Site not responding. Last check: 2007-10-07)
		As a corollary we observe that a sequence of degenerate groups converging algebraically on the boundary of a Bers' slice to a geometrically finite group does not converge strongly.
		We provide bounds on the exponent of convergence of a planar discrete quasiconformal group in terms of the associated dilatation and (a) the Hausdorff dimension of its conical limit set, or (b) the exponent of convergence of an underlying Kleinian group.
		In particular, we show that for a pair of geometrically isomorphic convex co-compact Kleinian groups, the ratio of the mass of the Patterson-Sullivan measure on line element space to the mass of its push-forward is bounded below by the ratio of the Hausdorff dimensions of the limit sets.
	ectaylor.web.wesleyan.edu /publist.html (1452 words)

	Kleinian group - Wikipedia, the free encyclopedia
		In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i.e.
		The group generated by inversion in each circle is a Kleinian group.
		The group of symmetries of the tessellation is a Kleinian group.
	en.wikipedia.org /wiki/Kleinian_group (430 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-07)
		The basic theory of Kleinian groups was laid down in the fundamental papers of H.
		The analytic approach to the theory of Kleinian groups is connected with the study of automorphic forms (cf.
		In particular, the theory of deformations of Kleinian groups, closely related to the theory of moduli of Riemann surfaces (see Moduli of a Riemann surface and Riemann surfaces, conformal classes of) relies on these methods.
	eom.springer.de /k/k055520.htm (852 words)

	BEING A KLEINIAN IS NOT STRAIGHTFORWARD
		Kleinians are thought to make long and penetrating interpretations based on what may appear to an unsympathetic outsider to be based on very little material.
		I can think of several people in the Kleinian world who have done truly admirable work, but the reverence in which they are held is really distressing and inconsistent with the advocacy of balance of the mixed nature of reality and sanity as characterised in the depressive position.
		Kleinianism is not straightforward, but it is profoundly helpful and therefore true in the sense of clinically fruitful.
	human-nature.com /rmyoung/papers/pap113h.html (7389 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-07)
		is a properly-discontinuous group of conformal automorphisms of
		In the uniformization of Riemann surfaces of finite type, the possible Kleinian groups may be classified.
		Teichmüller space) allows one to prove the possibility of simultaneous uniformization of several Riemann surfaces by a single Kleinian group, as well as that of all Riemann surfaces of a given type (cf.
	eom.springer.de /u/u095290.htm (888 words)

	Computational Kleinian Groups
		We were the first to produce computer graphics programs to draw limit sets of arbitrary free two-generator groups, and to map the space of these groups, as parametrized by the traces of their generators.
		Mas74], explained the calculation of the cusps on this boundary, showed the pattern of the `circle chain' in a cusp group, and gave a precise asymptotic formula for the shape of the cusps in the boundary.
		I gave this manuscript to Caroline Series when she visited Göttingen, and because of her expertise in the combinatorics of Kleinian groups and Riemann surfaces and ultimately three-manifolds she was the ideal person to prove and explain much of the patterns of circle chains that appear in the limit sets of cusp groups.
	www.math.okstate.edu /~wrightd/Works/research/node2.html (863 words)

	Introduction and statement of results
		For comparison recall the definition of quasiconformal stability from the theory of Kleinian groups as given by Bers in [2].
		As a consequence of this criterion, it follows that Fuchsian groups, Schottky groups, groups of Schottky type and certain non-degenerate B-groups are all quasiconformally stable [3].
		Because the image of a Schottky group under a homomorphism close to the identity is again a Schottky group of the same rank, the quasiconformal stability expressed by Theorem 1.1 includes quasiconformal stability in the sense of Bers.
	www.geom.uiuc.edu /~rminer/talks/cecm/latex2html/node1.html (840 words)

	Klein four-group - Wikipedia, the free encyclopedia
		In 2D it is the symmetry group of a rhombus and of a rectangle, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.
		The three elements of order 2 in the Klein four-group are interchangeable: the automorphism group is the group of permutations of the three elements.
		The Klein four-group is the group of components of the group of units of the topological ring of split-complex numbers.
	en.wikipedia.org /wiki/Klein_four-group (380 words)

	Petra Bonfert-Taylor's publications (Site not responding. Last check: 2007-10-07)
		We study discrete quasiconformal groups with small dilatation (that is dilatation close to 1) in n dimensions, n at least 3.
		We establish that a quasiconformal group is of compact type if and only if its limits set is purely conical and find that the limit set of a quasiconformal group of compact type is uniformly perfect.
		Convergence groups, Hausdorff dimension, and a Theorem of Sullivan and Tukia, (with J. Anderson and E.C. Taylor), Geometriae Dedicata 103 (2004), pp.
	pbonfert.web.wesleyan.edu /publist.html (1204 words)

	Spectral geometry and group cohomology (Site not responding. Last check: 2007-10-07)
		This group gives rise to a locally symmetric space of (in general) infinite volume, a decomposition of the geodesic boundary of this symmetric space into a limit set and a domain of discontinuity, and to a Selberg zeta function.
		Motivated by Patterson's conjectural description of the singularities of the Selberg zeta function we are interested in the cohomology of the discrete group with coefficients in these representations.
		Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group,
	www.uni-math.gwdg.de /bunke/project1.html (364 words)

	Geometry and Topology, Volume 4 (2000)
		We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface.
		Our main results give necessary conditions for the Kleinian group to have ‘bounded geometry’ (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves.
		Kleinian group, ending lamination, complex of curves, pleated surface, bounded geometry, injectivity radius
	www.msp.warwick.ac.uk /gt/2000/04/p003.xhtml (142 words)

	The Modular Group and Fractals
		The modular group doesn't just lead to Pellian equations and algebraic numbers, it in fact intertwines all rational numbers (and their extensions to reals and p-adics) in crazy, fractal ways.
		This paper provides the core background material for the structure of the Modular Group that is used in the other papers of this series.
		The goal here is to establish that these are once again given by the modular group, and that furthermore, these have utility in that they are automorphisms of the unit interval: they are both injective and surjective maps of the unit interval.
	www.linas.org /math/sl2z.html (1772 words)

	The Enneagram Institute Discussion Board - Object Relations basics...
		The theory holds that the infant's experience in relationship with the mother is the primary determinant of personality formation and that the infant's need for attachment to the mother is the motivating factor in the development of the infantile self.
		Although Independent and Kleinian theories are quite distinct from each other, and differ in the ways that each diverges from Sigmund Freud's theory of the mind, they are similar in focusing on the importance of the infant's experience of the mothering relationship.
		The Independent group followed Freud's use of the psychosexual stages of development, but disagreed that the inevitable progression of these stages was based on the primacy of the instincts.
	www.enneagraminstitute.com /forum/topic.asp?TOPIC_ID=1811 (4435 words)

	CMFT 2 (2002), 249--256 (Site not responding. Last check: 2007-10-07)
		A \emph{spherical point} of a Kleinian group $\Gamma$ is a point of $\mathbb{H}^3$ that is stabilized by a spherical triangle subgroup of $\Gamma$.
		An elliptic element of a Kleinian group is \emph{simple} if the translates of its axis under the group~$\Gamma$ form a disjoint collection of hyperbolic lines.
		We are also able to present substantial progress to the problem of identifying the minimal covolume Kleinian group.
	www.heldermann.de /CMF/CMF02/cmf0212.htm (207 words)

	topology seminars
		Suppose that $A$ and $B$ are two quasi-Fuchsian subgroups of a cocompact Kleinian group and which have intersection $C.$ Then there are subgroups A' and B' of finite index in A and B so that the subgroup generated by A' and B' is the amalgamated free product A' *_C B'.
		Thompson's group V is often described as the group of dyadic homeomorphisms of the Cantor set.
		Although most geometric group theorists (and low-dimensional topologists) have run across various combinatorial versions of the Gauss-Bonnet theorem for surfaces, fewer are aware of its natural generalization to arbitrary 2-complexes.
	www.math.ucsb.edu /~cooper/seminar/topseminar02to03.htm (2188 words)

	Spaces of Kleinian Groups and Hyperbolic 3-Manifolds
		The simplest class of Kleinian groups is the geometrically finite ones for which there is a finite sided fundamental polyhedron; the deformation theory of these groups was worked out by Ahlfors, Bers and Marden and largely understood pre-Thurston.
		Another conjecture of Thurston is that a Kleinian group is determined by the geometry of its convex core boundary.
		This will be a companion volume to Kleinian Groups and Hyperbolic 3-Manifolds, the Proceedings of the Warwick workshop, September 2001, which has recently appeared as LMS Lecture Notes 299 (2003).
	www.newton.cam.ac.uk /reports/0304/skg.html (1801 words)

	Kleinian Group Encyclopedia Article @ ChaosAndFractals.com (Chaos and Fractals) (Site not responding. Last check: 2007-10-07)
		orientation-preserving transformations are all isomorphic to the matrix group
		inversion in each circle is a Kleinian group.
		Fractals - Limit sets of kleinian groups, (undated) (links and additional references).
	www.chaosandfractals.com /encyclopedia/Kleinian_group (389 words)

	Introduction and statement of results
		The group of holomorphic isometries of [tex2html_wrap_inline673] is the Lie group [tex2html_wrap_inline675].
		Because the image of a Schottky group under a homomorphism close to the identity is again a Schottky group of the same rank, the quasiconformal stability expressed by Theorem
		As a corollary, Schottky groups of equal rank n (acting on [tex2html_wrap_inline831], [tex2html_wrap_inline833]) are quasiconformally conjugate.
	www.geom.uiuc.edu /~rminer/talks/cecm/webeq/node1.html (867 words)

	Kleinian and Fuchsian groups (via CobWeb/3.1 planetlab1.netlab.uky.edu) (Site not responding. Last check: 2007-10-07)
		An interesting property of Kleinian groups is that their limit sets have at most two points or are uncountable (a similar property holds for iteration of rational functions on the Riemann sphere).
		A Kleinian group whose limit set consists of at most two points is called an elementary group.
		For example the group consisting of all Möbius transformations with entries in the Gaussian integers, the Picard group, is discrete, but it does not act properly discontinuously at any point of the Riemann sphere.
	www.math.tifr.res.in.cob-web.org:8888 /~pablo/teichmuller/node5.html (717 words)

	Mathematical Sciences Research Institute - Teichmuller Theory and Kleinian Groups (Site not responding. Last check: 2007-10-07)
		The MSRI program in Teichmüller theory and Kleinian groups will address the need to strengthen connections between these two fields, and reassess new directions for each at a critical time in its history.
		The recent solutions of the tameness conjecture, density conjecture and the ending lamination conjecture put the study of hyperbolic 3-manifolds and Kleinian groups at a transitional point.
		More generally, there have been recent breakthroughs in understanding the extent of the analogies between the mapping class group and Kleinian groups, and the connections to Veech surfaces and the geometry of the mapping class group makes this area one of particular intererst for researchers in flows on moduli space and in hyperbolic geometry alike.
	www.msri.org /calendar/programs/ProgramInfo/246/show_program (344 words)

	Papers
		Ruelle proved that for quasiconformal deformations of cocompact Fuchsian groups the Hausdorff dimension of the limit set is an analytic function of the deformation.
		If $G$ is any Kleinian group we show the dimension of the limit set $\Lambda$ is always equal to either the dimension of the bounded geodesics or the dimension the geodesics which escape to infinity at linear speed.
		I show that a Kleinian group is geometrically finite iff its limit set consists entirely of conical limit points and parabolic fixed points.
	www.math.sunysb.edu /~bishop/papers/papers.html (6573 words)

	Papers
		My main areas of research are Hyperbolic geometry, and Kleinian groups.
		distortion of the Patterson-Sullivan measure of the Kleinian group.
		Bounds on the average bending of the convex hull of a Kleinian group
	www2.bc.edu /~bridgem/papers.html (223 words)

	Unfree Associations: Inside Psychoanalytic Institutes, CHAPTER 4, by Douglas Kirsner
		The new group was viewed as minimising the distinctions between psychoanalysis and psychotherapy,22 and was accused of 'sloppy' teaching by the classical group.23 Those influenced by Alexander were in favour of fewer hours per week for a training analysis.
		The new group viewed the old group as intolerant, stultifying and as a threat to academic freedom'.25 The increasing polarisation had virtually led to administrative and institutional paralysis26 and LAPSI was headed for a split.
		Unfortunately, rumors were circulated by both groups to the effect that the American Psychoanalytic Association considered the basic problem in the Los Angeles Institute to be the controversy in regard to Freudian and Kleinian points of view; this was not correct.
	www.academyanalyticarts.org /kirsner4.htm (16923 words)

	Limit sets of kleinian groups (Site not responding. Last check: 2007-10-07)
		Kleinian Groups by Jos Leys - g³ównie obrazki ale jakie !
		Drawing Limit Sets of Kleinian Groups Using Finite State Automata (1994) Greg McShane, John R. Parker, Ian Redfern.
		We have implemented these ideas in a C program, which we used to generate all the plots of limit sets shown here.
	republika.pl /fraktal/kleinian.html (246 words)

	Kleinian Group Fractals Main Page
		Kleinian group fractals have been popularized by the book "Indra's Pearls" by David Mumford, Caroline Series and David Wright.
		The key to fractals of this type is an understanding of Möbius transformations.
		Two examples of Kleinian group fractals are shown below:
	www.hiddendimension.com /KleinianGroup_Fractals_Main.html (293 words)

	Subsequent Years (Site not responding. Last check: 2007-10-07)
		The main objective of this seminar is to facilitate a discussion of some of the clinical differences between the contemporary British Kleinian and the American ego - psychological approaches to psychoanalysis.
		Participants are provided an introduction to the writings of selected contemporary British Kleinian analysts and the clinical thinking, in general, of this group.
		Clinical material provided by participants will be discussed from the two theoretical and technical perspectives, assisted by selected writings from the contemporary Kleinian group.
	www.bwanalysis.org /subsequent_years.htm (311 words)

	Page 2 for Paul McCreary's Home Page (Site not responding. Last check: 2007-10-07)
		The Isometric spheres of a Kleinian group bound the group's fundamental polyhedron.
		Cutting and pasting a surface hyperbolic polygon can have the same affect as applying the Mapping Class group to the surface which the polygon represents.
		Poincare described the conditions necessary and sufficient for a hyperbolic polygon to represent a Reimann surface.
	new.math.uiuc.edu /~paulmcc/KleinGrps.html (234 words)

	Melanie Klein and Critical Social Theory (Site not responding. Last check: 2007-10-07)
		From my initial scan of his work, I think that Professor Alford’s reading of Klein may be a little eccentric in parts for the British Kleinian; nevertheless it is an impressive conceptual investigation for which Kleinian thought is an unusual, and hitherto underused, instrument.
		Alford contends that a Kleinian account might solve problems which the Frankfurt School addressed but could not resolve because of their adherence to Freud’s notion of eros rather than the Kleinian nexus of love-hatred.
		The high point of the book is the formulation of a Kleinian group psychology which contrasts the intimacy of private relations with the aggressive-manipulative quality of group life, in a manner reminiscent of Niebuhr’s Moral Man and Immoral Society.
	yalepress.yale.edu /yupbooks/reviews.asp?isbn=0300105584 (636 words)

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