The Torelli geometry and its applications

By Benson Farb, Nikolai V. Ivanov. 2003

71 citations recorded by the source corpus.

Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod±(S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides a nondegenerate, skew-symmetric, bilinear form on H = H1(S,Z), called the intersection form. The natural action of Mod(S) on H preserves the intersection form. If we fix a symplectic basis in H, then we can identify the group of symplectic automorphisms of H with the integral symplectic group Sp(2g,Z) and the action of ModS on H leads to a natural surjective homomorphism

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/615063e6dc56307eb2e45401358c154c9de4c18c

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Citations in the corpus

Injections of Artin groups (2005, unknown): “method has been used to prove similar theorems by Korkmaz [27], Ivanov and McCarthy [23], Schmutz Schaller [31], Irmak [18] [17] [16], Margalit [28], Irmak, Ivanov, and McCarthy [19], Farb and Ivanov [12], McCarthy and Vautaw [29], Brendle and Margalit [5], and Irmak and Korkmaz [20]. In particular,”
Commensurations of the Johnson kernel (2004, background): “The parities of the ranks in Proposition 2.1 are crucial in Proposition 3.8.”
Curve Complexes and Finite Index Subgroups of Mapping Class Groups (2005, methodology): “Farb and Ivanov showed that the Torelli group, the subgroup of Mod(S) acting trivially on the homology of S, is co-Hopfian [10].”
A note on the connectivity of certain complexes associated to surfaces (2006, methodology): “We will give a simple, unified proof of the following theorem,which for C (Σg) and Cnosep(Σg) is due to Lickorish [13] (though he did not use this language) and forCsep(Σg) is due to Farb and Ivanov [3].”
COMBINATORIAL RIGIDITY IN CURVE COMPLEXES AND MAPPING CLASS GROUPS (2005, background): “Some of its other subgroups, in particular the Johnson kernel and the Torelli group, are of wide interest; see, respectively, [Brendle and Margalit 2004] and [Farb and Ivanov 2005], and references contained therein.”