Commensurations of the Johnson kernel

By Tara E. Brendle, Dany Margalit. 2004

62 citations recorded by the source corpus.

Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K) � Aut(K) � Mod(S). More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/34d2d489f2ee78576ea3b19483ebd7dd97eed6db

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Citations in the corpus

The Torelli geometry and its applications (2003, methodology): “Brendle-Margalit then built on our methods to extend the much harder second result to genus g ≥ 4; indeed they were even able to compute Comm(Kg) for the Johnson kernel - see [ BM ].”
Injections of Artin groups (2005, result): “His method has been used to prove similar theorems by various other authors [26, 28, 18, 20, 19, 15, 31, 8].”
Injective simplicial maps of the arc complex (2006, background): “hat an injection from a ﬁnite index subgroup of Kto the Torelli group, where Kis the subgroup of mapping class group generated by Dehn twists about separating curves, is induced by a homeomorphism in [BM1], [BM2]. Shackleton proved that injective simplicial maps of the curve complex are induced by h”
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.”
Curve Complexes and Finite Index Subgroups of Mapping Class Groups (2005, methodology): “Brendle–Margalit proved that the so-called Johnson kernel and all of its finite index subgroups are co-Hopfian [7]; this result and the previous were proven for closed surfaces of genus at least 4.”
Some problems on mapping class groups and moduli space (2006, background): “Brendle-Margalit [BM] built on [FI] to prove the harder results on Kg.”