the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed...

subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain...

explicit matrices. The mapping class group of a genus 2 surface is also known to be linear. In some cases the fundamental group of a manifold can be shown...

automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface. Jakob Nielsen (1924) showed...

problem is a question asked by Jakob Nielsen (1932, pp. 147–148) about whether finite subgroups of mapping class groups can act on surfaces, that was answered...

homeomorphisms of orientable surfaces of genus ≥ 2, but the type of a homeomorphism only depends on its associated element of the mapping class group Mod(S)....

of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface...

in the study of the mapping class group. Non-compact surfaces are more difficult to classify. As a simple example, a non-compact surface can be obtained...

topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mapping class group of a surface. The...

generators) for the mapping class group of a surface. The 3-manifold is the one that uses the word as the attaching map for a Heegaard splitting of the 3-manifold...

turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups. It was introduced...

quotient of Teichmüller space by the mapping class group. In this case it is the modular curve. In the remaining cases, X is a hyperbolic Riemann surface, that...

group is important in the topology of surfaces because there is a connection provided by the Dehn–Nielsen theorem: the extended mapping class group of...

a. It is a theorem of Max Dehn that maps of this form generate the mapping class group of isotopy classes of orientation-preserving homeomorphisms of...

are a special case of mapping tori. Here is the construction: take the Cartesian product of a surface with the unit interval. Glue the two copies of the...

She has made contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical...

for mapping class groups of closed surfaces. Nielsen transformations were introduced in (Nielsen 1921) to prove that every subgroup of a free group is...

space of a closed surface of genus g {\displaystyle g} is homeomorphic to a sphere of dimension 6 g − 7 {\displaystyle 6g-7} . The action of the mapping class...

particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically...

_{\operatorname {TOP} }(T^{n})\to 1.} The mapping class group of higher genus surfaces is much more complicated, and an area of active research. The torus's Heawood...

Other topics of interest Chiral knot Conjugacy problem Freiheitssatz Group isomorphism problem Lotschnittaxiom Mapping class group of a surface Non-Archimedean...

to" Mapping class group – Group of isotopy classes of a topological automorphism group Permutation group – Group whose operation is composition of permutations...

(2004). "On the action of the mapping class group for Riemann surfaces of infinite type". Journal of the Mathematical Society of Japan. 56 (4): 1069–1086...

PMID 17764440, S2CID 17402424 Mosher, Lee (1996), "A user's guide to the mapping class group: once-punctured surfaces", in Baumslag, Gilbert (ed.), Geometric and...

group: this reflects the fact that the mapping class group of a Riemann surface—the group of quasiconformal self-homomorphisms of a Riemann surface H/Γ...

representations of the mapping class groups of all closed surfaces. Associated to every modular tensor category C {\displaystyle {\mathcal {C}}} , it is a theorem...

Liouville's theorem sharply limits the conformal mappings to a few types. The notion of conformality generalizes in a natural way to maps between Riemannian or...

Hyperbolic groups Mapping class groups (automorphisms of surfaces) Symmetric groups Braid groups Coxeter groups General Artin groups Thompson's group F CAT(0)...

Roman surface Steiner surface Alexander horned sphere Klein bottle Mapping class group Dehn twist Nielsen–Thurston classification Moise's Theorem (see also...

only be specified up to taking a double coset in the mapping class group of H. This connection with the mapping class group was first made by W. B. R. Lickorish...