
	Where results make sense

Topic: Fuchsian group

Related Topics

Discrete group

SL(2,Z)

Kleinian group

Projective linear group

Fuchsian model

Kleinian model

Hyperbolic geometry

Isometry

Riemann integration

Riemannian geometry

Euclidean space

Fibonacci number

Open ball

Spherical geometry

Riemannian manifold

	Fuchsian group - Wikipedia, the free encyclopedia
		In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane.
		A Fuchsian group is always a discrete group, and thus is a lattice in one of the two semisimple Lie groups PSL(2,R) or PSL(2,C).
		All hyperbolic and parabolic cyclic subgroups of PSL(2,R) are Fuchsian.
	en.wikipedia.org /wiki/Fuchsian_group (1087 words)

	Fuchsian group - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-08)
		A Fuchsian group is always a discrete group, and is a special case of a lattice in a semisimple Lie group.
		Fuchsian groups are especially important in geometry, number theory, and the theory of dynamical systems.
		This action is faithful, and in fact G is isomorphic to the group of all orientation-preserving isometries of H.
	encyclopedia.worldsearch.com /fuchsian_group.htm (410 words)

	Discrete group - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-08)
		In mathematics, a discrete group is a group G equipped with the discrete topology.
		Since every map from a discrete space is continuous, the topological homomorphisms of a discrete group are exactly the group homomorphisms of the underlying group.
		Hence, there is an isomorphism between the categories of groups and of discrete groups and indeed, discrete groups can generally be identified with the underlying (non-topological) groups.
	encyclopedia.worldsearch.com /discrete_symmetry_group.htm (321 words)

	Fuchsian model - Wikipedia, the free encyclopedia
		In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R.
		The quotient space H/Γ is then a Fuchsian model for the Riemann surface R.
		The Nielsen isomorphism theorem basically states that the algebraic topology of a closed Riemann surface is the same as its geometry.
	en.wikipedia.org /wiki/Fuchsian_model (239 words)

	Talk:Fuchsian group - Wikipedia, the free encyclopedia
		However, the page gives the modular group PSL(2,Z) as an example of a Fuchsian group, and the modular group is not torsion free.
		Well, no, Fuchsian group is specific, as is Kleinian group, Picard group.
		The lattice (group) definition has an advantage, namely it says the quotient space G/Γ has finite invariant measure; and this is something that passing to a quotient by a compact subgroup respects.
	en.wikipedia.org /wiki/Talk:Fuchsian_group (867 words)

	Fuchsian group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Fuchsian groups are used to create (Click link for more info and facts about Fuchsian model) Fuchsian models of (Click link for more info and facts about Riemann surface) Riemann surfaces.
		This action is faithful, and in fact G is isomorphic to the group of all (Click link for more info and facts about orientation-preserving) orientation-preserving (Click link for more info and facts about isometries) isometries of H.
		A Fuchsian group is also (Click link for more info and facts about torsion free) torsion free, meaning it has no finite (Click link for more info and facts about abelian) abelian subgroups.
	www.absoluteastronomy.com /encyclopedia/f/fu/fuchsian_group.htm (408 words)

	Edward Taylor's Publications (Site not responding. Last check: 2007-10-08)
		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.
		We show that a discrete, quasiconformal group preserving n-dimensional hyperbolic space has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity.
	ectaylor.web.wesleyan.edu /publist.html (1373 words)

	Modular curve - Wikipedia, the free encyclopedia
		In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as
		where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices.
		that are compact do occur with Fuchsian groups Γ other than subgroups of the modular group; these are of interest in number theory, also, in cases where they are constructed from quaternion algebras.
	en.wikipedia.org /wiki/Modular_curve (380 words)

	Alternating Quotients Of Fuchsian Groups - Everitt (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		This settles in the affirmative a long-standing conjecture of Graham Higman.
		2 Subgroups of Fuchsian groups and finite permutation groups (context) - Singerman - 1970
		1 Permutation Representations of the Symmetry Groups of Regula..
	citeseer.ist.psu.edu /everitt00alternating.html (565 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.
		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.
		This is an expansion and revision of the old draft (2000) in the art gallery.
	www.linas.org /math/sl2z.html (1340 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The fundamental group $G$ of $M$ is the fundamental group of a graph $\Gamma$ of groups, whose underlying graph is dual to the frontier of $V(M)$.
		Thus the edge groups of $\Gamma$ are all isomorphic to $\mathbb{Z}$ or $\mathbb{Z}\times\mathbb{Z}$, and the vertex groups are the fundamental groups of simple manifolds or of Seifert fibre spaces or of surfaces.
		As Fuchsian groups play an important role in the results, we should explain that we use the term to include not only discrete groups of isometries of the hyperbolic plane, but also to include discrete groups of isometries of the Euclidean plane.
	www.univie.ac.at /EMIS/journals/ERA-AMS/2002-01-003/2002-01-003.tex.html (3336 words)

	CONFORMAL ANALYSIS AND GEOMETRY (Site not responding. Last check: 2007-10-08)
		The main areas of the research of the team are (1) analysis on manifolds and on metric spaces, (2) Beurling operators, (3) Möbius groups and their generalizations, and (4) quasiconformal and quasiregular mappings.
		In (3) the central theme is to study the limit behavior of a Fuchsian group while deformed along a geodesic in a Teichmüller space.
		The purpose of the group is to build a complete BLD theory on generalized manifolds with controlled mass and rectifiability.
	www.helsinki.fi /~iholopai/conage.html (205 words)

	Brent Everitt's Research
		The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic n-manifolds.
		It is shown that any Fuchsian group has among its homomorphic images all but finitely many of the alternating groups A_n.
		In this paper we show how to obtain representations of Coxeter groups acting on H^n to certain classical groups G. We determine when the kernel K of such a homomorphism is torsion-free and thus H^n/K is a hyperbolic n-manifold.
	www-users.york.ac.uk /~bje1/research.html (752 words)

	abstract Shalev 2003 (Site not responding. Last check: 2007-10-08)
		Fuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts.
		They include surface groups, triangle groups, the modular group, free groups, etc. We use character-theoretic and probabilistic methods to study spaces of homomorphisms from Fuchsian groups to symmetric groups and to finite simple groups.
		We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as mixing times of random walks on symmetric groups and simple groups.
	www.win.tue.nl /~amc/seminar/abstracts/shalev.html (135 words)

	Colloquium Announcement (Site not responding. Last check: 2007-10-08)
		Abstract: We define the average bending of a geodesic on the boundary of the convex hull of a quasi-fuchsian group and prove that it is universally bounded by a number $K$.
		We use this to prove that $1+K$ is a universal bound on the lipschitz constant for the map from the intrinsic hyperbolic structure on the convex hull boundary to the hyperbolic structure on the domain of discontinuity facing it.
		We further prove that the length of the bending lamination of the convex hull of a quasi-fuchsian group is bounded by $K \pi^2$ times the euler characteristic of the underlying surface
	math.dartmouth.edu /~colloq/f00/2000-October-19_969553120.phtml?s=CIN (112 words)

	Geometry/Topology Seminar
		For the case of groups of C^1 diffeomorphisms one can consider the Lyapunov exponent with respect to the Lebesgue measure: I will give the ideas of the proof of the fact that this number is negative for non elementary actions.
		Abstract: Automorphism groups of fundamental groups of surfaces (such as "mapping class groups") are mysterious groups which arise in many mathematical contexts.
		These groups act on spaces of representations of surface groups, preserving natural symplectic or Poisson geometries and invariant smooth measures.
	www.math.uchicago.edu /~geometry (762 words)

	CCSD thèses-EN-ligne: Transformations hyperboliques et courbes algebriques en genre 2 et 3 (Site not responding. Last check: 2007-10-08)
		The uniformization theorem of Poincaré-Koebe states that any smooth compact Riemann surface of genus $g>1$ is a quotient of the upper half-plane by a Fuchsian group.
		We describe the correspondence between the actions of two groups, the first acting on the algebraic structures, and the second on the hyperbolic structures of these surfaces.
		We then study special algebraic families, in which the surfaces are defined by a smaller number of parameters than those of the ambient spaces (but not having necessarily more automorphisms).
	tel.ccsd.cnrs.fr /documents/archives0/00/00/11/54/index_fr.html (660 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Inversely, if a hyperelliptic Riemann surface is defined by a discontinuous group of M\"obius transformations of the second kind, say acting in the unit disc, then the algorithm computes an equation describing the surface as an algebraic curve.
		The group has been particularly interested in Riemann surfaces with a \lq\lq large" group, of automorphisms for their properties may often be determined by group-theoretic and combinatorial methods.
		The study of Riemann surfaces with large automorphism group is equivalent to the study of normal subgroups of Triangle groups.
	dmawww.epfl.ch /geometrie/activities.old/buser (2677 words)

	[No title]
		It describes the dynamics of the modular group action on characters of isometric actions of rank two free groups on hyperbolic 3-space which preserve a hyperbolic plane (but not necessarily an orientation on the hyperbolic plane).
		The main result of this paper, written jointly with Walter Neumann, is that the action of the modular group on the homology of the SL(2,C)-character varieties of a one-holed torus and a four-holed sphere factor through a finite group.
		This paper, coauthored with Todd Drumm, shows that the Margulis invariant is a complete invariant of the translational conjugacy class of an affine deformation of a Fuchsian group.
	www.math.umd.edu /~wmg/publications.html (1614 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		When the surface group for genus g is factored out of all of these, the inclusion lattice of automorphism groups for non-hyperelliptic Riemann surfaces of genus 3 is obtained.
		Generators and relation for these finite groups are found in terms of the generators and relations for the Fuchsian groups.
		The explicit inclusion maps are determined by a method which views the fundamental domain of a Fuchsian group of one type as included in the fundamental domain of a Fuchsian group of a second type.
	www.fsu.edu /gradstudies/thesis/1997/Fall97/xiaozhong.html (227 words)

	Patterson Measure And Ubiquity - Dodson, Meli, Pestana, Velani (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		By extending the notion of ubiquity from k -dimensional Lebesgue measure to m, a natural lower bound for the Hausdorff dimension of a fairly general class of lim sup subsets of L is obtained.
		This is applied to Patterson measure supported on the limit set of a convex co-compact group, to...
		0.3: The Exponent of Convergence of Kleinian Groups; on a Theorem of..
	citeseer.ist.psu.edu /dodson95patterson.html (478 words)

	Analysis-Geometry Seminar
		We further prove that the length of the bending lamination of the convex hull of a quasi-fuchsian group is bounded by K (pi)
		Abstract: An isometric action of a Lie group on a Riemannian manifold is called polar if there exists a submanifold which intersects all orbits orthogonally.
		Abstract: I will present a new generalized version of an index of a Dirac operator on a complete Riemannian manifold endowed with an action of a compact Lie group G. I show that this index is an invariant of a non-compact cobordism of the type considered by V. Guillemin, V.
	www.math.neu.edu /~mcowen/AGSeminar99-00.html (2096 words)

	[No title]
		Let $\G$ be a fuchsian subgroup of $PSL(2,\Bbb R)$, which has infinite conjugacy classes and is of infinite covolume.
		\proclaim{Proposition 3} Let $\G$ be a Fuchsian group of infinite covolume in $PSL(2,\Bbb R)$, with infinite conjugacy classes and let $\Cal A_t$ be the $II_\infty$ factor consisting of all bounded operators on $H_t$ that commute with $\pi_t(\G)$.
		We apply now the machinery from ([CP2]) to use the computation of traces of commutators to determine the index of a Toeplitz operator with $\G$-invariant symbol, that is invertible on the closure of the upper halfplane.
	www.math.uiowa.edu /~radulesc/paperNestFlorin (2132 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		To appear in Ann.\ Acad.\ Sci.\ Fenn.\ A.\ I. \title{Discrete groups and thin sets} \author{Torbj\"orn Lundh} \begin{abstract} Let $\Gamma$ be a discrete group of M\"obius transformations acting on and preserving the unit ball in $\Rdim$ (i.e.\ Fuchsian groups in the planar case).
		One of the answers that will be given says that the critical exponent of $\Gamma$ %(see page~\pageref{krit}) equals the Hausdorff dimension of the set on the unit sphere where the archipelago of $\Gamma$ is not minimally thin.
		Another answer tells us that the limit set of a geometrically finite Fuchsian group $\Gamma$ is the set on the boundary where the archipelago of $\Gamma$ is not rarefied.
	www.math.sunysb.edu /~tobbe/ar.abst (132 words)

	Modular curve (Site not responding. Last check: 2007-10-08)
		where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modulargroup of integral 2×2 matrices.
		Here for any N ≥ 1 Γ(N) is thesubgroup of the modular group of matrices that are in the kernel of reduction modulo N, and Γ
		that are compact do occur with Fuchsian groups Γ other than subgroups of the modular group; these are ofinterest in number theory, also, in cases where they are constructed from quaternion algebras.
	www.therfcc.org /modular-curve-79311.html (350 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		We introduce a new family of three dimensional quasi-fuchsian groups and the computer visualization of their limit sets as new fascinating fractal shapes which are natively embedded in the tree dimensional space.
		Very few examples of quaisfuchsian groups in three dimensional space are known, they had never been visualized.
		The limits set of fuchsian group can be defined as round sphere, and those of quasi-fuchsian group is just known as not round surfaces which topologically are equivalent to sphere, but no one knew how non-round they were.
	www.newton.cam.ac.uk /programmes/SKG/poster/araki.html (110 words)

	Discrete groups and the Dirichlet polygon (Site not responding. Last check: 2007-10-08)
		A group G is a topological group, if it has a topology such that the inverse and the group operation
		Figure 3 shows z=0 and those image points which are closest to z for a particular Fuchsian group.
		Figure 4 is an example of illustrating the Dirichlet polygon using the generators for a Fuchsian group.
	www.csc.fi /math_topics/DH/node5.html (191 words)

Try your search on: Qwika (all wikis)