In, jorgensen gave a complete description of the combinatorial structure of the ford domain of every quasi fuchsian oncepunctured torus group see also. Zariski density and finite quotients of mapping class groups, international mathematics research notices 20, no. Pdf we give a new realanalytic embedding of the teichmuller space of closed riemann surfaces of genus 2 into r6. Since every fuchsian group may be obtained in this way, we say that a fuchsian group is derived from a quaternion algebra if there exists a quaternion algebra and an order. Fuchsian codes for awgn channels universitetet i bergen. Heins, fundamental polygons of fuchsian and fuchsoid groups,ann. Her research specialties include complex geometry, fuchsian groups, riemann surfaces, and abelian varieties. The subject of fuchsian groups lies at the intersection between geometry, topology, algebra, number theory, and analysis, so it should be of interest to a wide range of graduate students, including rstyear students. Wolpert nsfcbms lectures, central connecticut state university july 2024, 2009 january 15, 2009 scott a. Past work has emphasized general relativity, studying model spacetimes and their properties, as well as the interface between relativity and quantum physics. He moved to jerusalem at 18 to study mathematics and philosophy at the hebrew university, and since then, excluding some years abroad, he has been living mainly in jerusalem shalev received his ph. Guest editor of special issue of the journal of algebra for 60th birthday of gordon james, december 2006. We shall call a hyperbolic element 7 reduced if cj intersects the given fundamental region ro see sect.
Hence the volumes of two pseudodeveloping maps for a given representation can be different. In chapter 2 he presents the appropriate automorphic functions, in particular, fuchsian functions. His doctoral thesis was written under the supervision of. Almost anything that your organisation might need to write or edit, including ads, annual reports, articles and blogs, brochures, case studies, emails and letters, media releases, newsletters, prospectuses, research reports, speeches, tenders and proposals, videos and audio scripts, websites and whitepapers. A very pleasant account of the whole story can be found in 6.
To learn about our use of cookies and how you can manage your cookie settings, please see our cookie policy. Download fulltext pdf on almost primitive elements of free groups with an application to fuchsian groups article pdf available in canadian journal of mathematics 452 april 1993 with 32 reads. The chicago distribution center will reopen for order fulfillment on april. A group g of isometries of the hyperbolic plane is said to be of type a. Topics in geometry dartmouth college winter 20 time tba this course will be a general introduction to fuchsian groups, with emphasis on spectral geometry and arithmetic groups. These studies lead us to explore infinite paths in suborbital graphs for the fuchsian group. Pdf on almost primitive elements of free groups with an. By closing this message, you are consenting to our use of cookies. Riemann surfaces of gen us 2, all fuchsian groups are pu rely hyperbolic, and all references to lengths, distances, etc. New parameters for fuchsian groups of genus 2 article pdf available in proceedings of the american mathematical society 12712 january 1999 with 40 reads how we measure reads. Basics symplectic geometry curvature metric geometry. Editorial board, 20142018 msri publication book 61 thin groups and superstrong approximation, cambridge university press, coedited with breuillard.
He discusses the corresponding isometry groups, introduces the idea of discrete subgroups, and shows that the corresponding quotient spaces are manifolds. In this paper, we focus on a certain type of such groups, socalled pseudofibered groups, and show that many 3manifold groups are examples of pseudofibered groups. Riemann surfaces of genus 2, all fuchsian groups are purely. Thus we are concerned with a steady flow in the following concepts are of fundamental significance for the study of dynamical flows. Such groups must be discrete otherwise the quotient structure would not be satisfactory and these groups are the fuchsian groups. Topics include the theory of affine structures and polynomial structures, affine schottky groups and crooked tilings, theory and problems on the geometry of finitely generated solvable groups, flat lorentz 3manifolds and fuchsian groups, filiform lie algebras, hyperbolic automorphisms and anosov diffeomorphisms on infranilmanifolds. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the opposite case, o2 oo, we speak of the groups and surfaces of the second kind. Writing tr ax, tr by and tr ab2 one sees that the condition on.
History of mathematics volume 10 sources of hyperbolic. Explicit bounds on automorphic and canonical green. Fuchsian groups by svetlana katok, 9780226425832, available at book depository with free delivery worldwide. Irkutsk, 1989, l a bokut, m hazewinkel, y g reshetnyak, editors, amer. Hadamard begins with hyperbolic geometry, which he compares with plane and spherical geometry. Maximal volume representations are fuchsian note that vold and hence vol. The princeton companion to mathematics editor timothy gowers university of cambridge associate editors june barrowgreen. The indices of torsionfree subgroups of fuchsian groups authors. We study limits of quasifuchsian groups for which the bending measures on the convex hull boundary tend to zero, giving necessary and sufficient conditions for the limit group to exist and be. We use cookies to give you the best possible experience. Out of curiosity, i have given a try to several proof assistants, i. Faculty research mathematics oregon state university. A quasifuchsian manifold with the associated convex core.
A kleinian group is a discrete group of isometries of hyperbolic 3space h3. Topics addressed in the book include amenable groups, kaehler groups, automorphism groups of rooted trees, rigidity, calgebras, random walks on groups, prop groups, burnside groups, parafree groups, and fuchsian groups. These previous concepts and results lead to the concept of arithmetic fuchsian groups. Anexample ofan application ofthesecond method isthetiling ofbricks thatone gets when one builds a wall. Weilpetersson sampler a highlight of themes, current understanding and research scott a. In the hyperbolic plane, which is the richest source for riemann surfaces, these groups are called fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. Infinite paths of minimal length on suborbital graphs for. Epstein, editor, analyti cal and geometric aspects of. Margulis, coedited with burger, ji, lubotzky and yau. Weil, we know that all arithmetic subgroups of sl2. This work has not been previously formed as the basis for the. Abstractfollowing previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations.
Our results are valid for any cofinite fuchsian group, although we are motivated by the case of arithmetic groups, and in particular congruence. Equivalently, two branched projective structures with the same quasifuchsian holonomy and the same number of branch points. Fuchsian groups, chicago lectures in mathematics, university of chicago press, 1992. Kleins riemann surface with 168 symmetries is a beautiful example. For the academic year 19801981 he was a guggenheim fellow. With so many projects on hold, it can be difficult to know what to do with yourself and your team each day. R, up to a relation called commensurability, are given by fuchsian groups derived from quaternion algebras over totally real number. Computations of eisenstein series on fuchsian groups 1781 gaussian value distribution as ims. Exotic components in linear slices of quasifuchsian groups. Topologically singular points in the moduli space of riemann surfaces. The subject of fuchsian groups lies at the intersection between geometry, topology, algebra, number. If the address matches an existing account you will receive an email with instructions to reset your password.
Received by the editors october 20, 1997 and, in revised form, february 20. Large symmetry tessellations of the hyperbolic plane are just as fascinating. Fuchsian groups and ergodic theory f by eberhard hopf introduction. Shimura curves with many uniform dessins 3 1 the main question and the answer in the easy case to put the problem in a precise form we observe.
Arithmetic fuchsian groups arising from quaternion algebras. One of the important features of those groups lies by the concept of uniformization of surfaces. Liebeck appointment professor of pure mathematics, imperial college london. C and 2 are of the first kind if the surface area of 2 or, what is the same, the nearea of the fundamental region r is finite. Papers dedicated to professor mikio sato on the occasion of his 60th birthday, volume ii is a collection of research papers on algebraic analysis and related topics in honor to professor mikio satos 60th birthday. Topics on riemann surfaces and fuchsian groups pdf free. Daoyin, the traditional chinese practice of guiding the qi and stretching the with historical context and. Otherwise, a fuchsian group is said to be of the second type. Pdf limits of quasifuchsian groups with small bending. Codes over graphs derived from quotient rings of the.
Discrete subgroups of psl2,r are usually called fuchsian groups. Any hyperbolic 3manifold is the quotient of h3 by a kleinian group. It is natural to expect that there is an analogue of jorgensens theory for quasi fuchsian oncepunctured klein bottle groups, because its deformation space has also complex dimension 2. Katok founded the electronic research announcements of the american mathematical society in 1995. Mg consists of the surfaces of genus g having a subgroup of their automorphism groups isomorphic to g with a speci. On the arithmetic group we cannot see resemblance ofacomplexgaussian distribution whenres12duetoaclustering. London mathematical society lecture note series managing editor. Examples appear in number theory modular groups and triangle groups, the theory of elliptic functions, and the theory of linear differential equations in the complex domain giving rise to the alternative name fuchsian groups. In, yayenie gave a remark stating that the fuchsian group acts transitively on the set if and only if m is either 1 or prime. Bibliography of david gabai 1 the murasugi sum is a natural geometric operation, contemporary mathematics 20 1983, 1145. Lectures on fuchsian groups and their moduli institut fourier. Ams transactions of the american mathematical society. University uaeu, and the author of this thesis entitled on the geometry of fuchsian groups, hereby, solemnly declare that this thesis is an original research work that has been done and prepared by me under the supervision of dr.
A fundamental domain for some quasifuchsian groups takehiko sasaki received november 28, 1988 revised april 1, 1989 1. On laminar groups, tits alternatives and convergence group. Discontinuous groups of isometries in the hyperbolic plane. We then prove that torsionfree pseudofibered groups satisfy a. Other past and present interests include algebraic computing, asymptotic structure, noneuclidean. A fuchsian group of the first type is a group for which the limit set is the closed real line r. Current work investigates applications of the octonions, the unique nonassociative division algebra, to the physics of fundamental particles. Theory of fuchsian groups, memoir on kleinian groups, on the applications of noneuclidean geometry to the theory of quadratic forms 1 translation of poincares theory of fuchsian groups 123 translation of poincares memoir on kleinian groups 1 translation of poincares on the applications of noneuclidean geometry to the theory of. Introduction let a and b be two loxodromic elements of sl2, c having no common fixed point and subjected to the condition that aba1 b. In, jorgensen gave a complete description of the combinatorial structure of the ford domain of every quasifuchsian oncepunctured torus group see also. It contains survey papers contributed by leading specialists in group theory and other areas of mathematics. Received by the editors august 1, 1982 and, in revised form, february 3, 1983. Canonical rings, canonical embeddings, stacks, algebraic curves, modular forms, automorphic forms, generic initial ideals, gr obner bases. At editor group, were thinking about how we can help our clients by adapting our services for the new environment especially our writing training.
In this lecture we will construct the arithmetic fuchsian groups in terms of quaternion algebras. The canonical ring of a stacky curve dartmouth college. A century of mathematics in america, part ii 1989 1 peter l. Bujalance et al 288 surveys in combinatorics, 2001, j. Canonical polygons for finitely generated fuchsian groups. Among important classes of fuchsian groups are those known as the triangle groups. He is a past co editor of connecting research to teaching for nctms mathematics teacher journal, associate editor for school science and mathematics, an editorial panel member for the journal for research in mathematics education, and an editorial panel member for the mathematics teacher educator. A kleinian group is a discrete group of isometries of hyperbolic 3 space h3. Theory of fuchsian groups, memoir on kleinian groups, on the applications of noneuclidean geometry to the theory of quadratic forms 1 translation of poincares. Co editor of the special issue of quarterly journal of pure and applied math. This is an introductory textbook on isometry groups of the hyperbolic plane.
Reduction theory for fuchsian groups 465 invariant. This is in contrast to the cases of fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. Interest in such groups dates back more than 120 years. Philosophical transactions of the royal society of london. Introduction it is wellknown that the set of quasisymmetric functions does not form a topological group in the quasisymmetric topology cf. We suppose that every motion can be continued along the entire timeaxis. Chicago lectures in mathematics the chicago distribution center will reopen for order fulfillment on april. Hitchin, mathematical institute, university of oxford, 2429 st giles.
Asymptotically rigid mapping class groups and thompson groups with christophe kapoudjian and vlad sergiescu handbook of teichmuller theory vol iii editor a. It is natural to expect that there is an analogue of jorgensens theory for quasifuchsian oncepunctured klein bottle groups, because its deformation space has also complex dimension 2. Shalev was born in kibbutz kinneret and grew up in beit berl. Algorithmic construction of fundamental polygons for certain. Cj becomes a closed geodesic in f\d and it can be coded according to the order it intersects the sides of r o.
Since any finite subset of a metric space is discrete. Noneuclidean geometry in the theory of automorphic functions. This course will be a general introduction to fuchsian groups, with em phasis on spectral geometry and arithmetic groups. The book series chicago lectures in mathematics published or distributed by the university of chicago press. Algorithmic construction of fundamental polygons for. These groups form a slice in quasifuchsian space, homeomorphic to the teichmulle. Fundamental polygons for fuchsian groups springerlink. We prove that if s is a closed compact surface of genus g.
So, we study on the fuchsian group where m is a prime number. Character values and fuchsian groups, 20042007 unipotent elements of algebraic groups, 20072010. I continue to be fascinated by the beauty and intricacy of riemann surfaces, especially descriptions by fuchsian groups, by cutandpaste constructions and as solutions of algebraic equations. Mg consists of the surfaces of genus g having a subgroup of their automorphism groups isomorphic to g with a. Hejhal list of publications 1 on convergence properties of sequences of analytic functions, american mathematical monthly 741967, 12081211. It is immediate that because in every case the group f also acts as a group of isometrics of the underlying geometry, the geometric structure passes down under the quotient map to the riemann surface. Ford domains of fuchsian oncepunctured klein bottle groups. All chicago ebooks are on sale at 30% off with the code ebook30. The book fuchsian groups, svetlana katok is published by university of chicago press.