On the displacement of generators of free fuchsian groups. The papers i have selected include the three ileat memoirs in the first volumes of acta math. They are particular instances of arithmetic groups. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the twodimensional margulis constant and lengths of closed curves. A basic example of lattices in semisimple groups, fuchsian groups have extensive connections to the theory of a single complex variable, number theory, algebraic. See 6, chapters 24, or 9, chapters 12, for succinct expositions of the basic theory of fuchsian groups. Intro david sprunger 1 topological groups in various ways, mathematicians associate extra structure with sets and then investigate these objects. In this paper we show that all fuchsian schottky groups are classical schottky groups, but not necessarily on the same set of generators.
Conder department of mathematics, university of auckland private bag 92019, auckland 1142, new zealand. Finitely generated fuchsian groups exercises for chapter 4 5. Arithmetic fuchsian groups of genus zero rice university math. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the twodimensional margulis. Fuchsian groups, finite simple groups and representation varieties. Jan 11, 2020 fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. Fuchsian groups, circularly ordered groups, and dense.
This yields a necessary condition for a quasi fuchsian group to be almost fuchsian which involves only conformal geometry. Definitely recommend it for selfstudy or teaching a first course in hyperbolic geometry. Jul 27, 2019 fuchsian groups by svetlana katok, available at book depository with free delivery worldwide. As an application, we prove that there are no doublydegenerate geometric limits of almost fuchsian groups. All fuchsian schottky groups are classical schottky groups. They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among fuchsian groups. Suppos fe is the free grou opn two generators a and b. This introductory text provides a thoroughly modern treatment of fuchsian groups that addresses. Definitely recommend it for selfstudy or teaching a. In discrete groups and geometry, papers dedicated to macbeath, a.
Reduction theory for fuchsian groups 465 invariant. In addition, if g is cocompact, then any of its associated. When the hyperbolic surface hg is compact, the fuchsian group g is called cocompact. We shall call a hyperbolic element 7 reduced if cj intersects the given fundamental region ro see sect. This yields a necessary condition for a quasifuchsian group to be almostfuchsian which involves only conformal geometry. Psl2r autch is the group of automorphisms of h as a com plex manifold. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the twodimensional margulis constant and lengths of closed curves, which. R be an embedding, o be an order of b, and o1 the elements of norm one in o. Torsion free subgroups of fuchsian groups 457 an alternate formulation is the following. Fuchsian groups, coverings of riemann surfaces, subgroup. A torsion free subgroup of finite index in a finitely generated fuchsian group is isomorphic to itm where m is an oriented surface of finite type. A fuchsian group is a group that acts properly discontinuously on the upper half plane. On the closeness of approach of complex rational fractions to a complex irrational number. Fuchsian groups, finite simple groups and representation.
Representations of free fuchsian groups in complex hyperbolic. Buy fuchsian groups chicago lectures in mathematics by svetlana katok isbn. Lectures on fuchsian groups and their moduli institut fourier. In the opposite case, o2 oo, we speak of the groups and surfaces of the second kind. A representation of fuchsian groups article pdf available in journal of algebra 1791. Note that foxs paper contains an error, corrected by chau 1. As an application, we prove that there are no doublydegenerate geometric limits of almostfuchsian groups. We estimate the dimension of varieties of the form homf,g where f is a fuchsian group and g is a simple real algebraic group, answering along the way a. Mathematical institute, university of oxford andrew wiles building, roq, woodstock road, oxford ox2 6gg, u.
A family ffp j p2 pg of subsets of a metric space mindexed. Topology 39 2000 3360 representations of free fuchsian groups in complex hyperbolic space nikolay gusevskii. Aug 06, 2019 fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. Torsion free noded fuchsian groups come in two flavors 20, 21. Examples are the fundamental groups of closed, hyperbolic surfaces. Complex length coordinates for quasifuchsian groups.
Representations of free fuchsian groups in complex. However, in 26, schick shows the conjecture is false for z4 z3, and other counterexamples have since been constructed in. For example, by adding a binary operation to a set and validating that the set with the binary operation ful lls certain axioms, we arrive at a group. In particular, nonabelian free groups are fuchsian, as well as free products of. Cj becomes a closed geodesic in f\d and it can be coded according to the order it intersects the sides of r o. Jul 14, 2019 katok fuchsian groups pdf fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. Fuchsian groups and tessellations of h a fuchsian group is a discrete group f of isometrics of the hyperbolic plane h the unit disk endowed with the poincar6 metric dn. Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. In nite groups for which the conjecture is proven are free and free abelian groups 25, certain other torsion free groups, and some crystallographic groups with prime torsion 7.
Cocompact fuchsian groups are also interesting early examples of groups with solvable word and conjugacy problem, using ideas related to dehns algorithm. Katok fuchsian groups pdf fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. The exercises are great and there are hints for the selected ones. Quasifuchsian groups recall that a fuchsian group of type i is a kleinian group. The group psl2,r acts on h by linear fractional transformations also known as mobius transformations. Topics on riemann surfaces and fuchsian groups london. Every triangular group is a subgroup of index 2 in the group generated by reflections relative to the sides of a triangle with angles, see reflection group. We may obtai an representatio onf f as a fuchsian group acting in the unit dis dk as follows. E0 n for n squarefree, which are familiar as the normalizers in psl2,r of the. Arithmetic fuchsian groups are a special class of fuchsian groups constructed using orders in quaternion algebras.
On the other hand, psl2r acts on h by holomorphic automorphisms, and we also have theorem 2. Fuchsian group which is commensurable with psl 2z is obtained by taking the normalizer n0n of 0n for nsquare free, where the normalizer is taken in psl 2r. This introductory text provides a thoroughly modern treatment of fuchsian groups that addresses both the classical material and recent developments in the field. Then g is a fuchsian group such that h2g has no cusps if. Therefore, all such groups are necessarily torsionfree. If x hk where k is a torsionfree fuchsian group one with no. Edmonds and others published torsion free subgroups of fuchsian groups and tessellations of surfaces find, read and cite all the research you need on researchgate. The indices of torsionfree subgroups of fuchsian groups. Fuchsian group acting on the unit disc d containing. The most interesting fuchsian groups are those with s t 0. These are the papers which made his reputation and they include many results and proofs which are now standard. A fuchsian group is a discrete subgroup of psl2, r.
Then the pingpong lemma shows that this group is free. This action is faithful, and in fact psl2,r is isomorphic to the group of all orientationpreserving. Cocompact fuchsian group together with finite rank free groups are the early examples of hyperbolic groups in the sense of gromov, proved using that the group with its word metric is quasi. Indeed, even the modular group psl 2, zwhich is a fuchsian group, does not act discontinuously on the real number line. A very precise, yet intuitive approaach to fuchsian groups.
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