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Let $Gamma$ be a finite index subgroup of the mapping class group $MCG(Sigma)$ of a closed orientable surface $Sigma$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $ginGamma$ has positive stable commutator length. In addition, we show that in these situations the stable commutator length, if nonzero, is uniformly bounded away from 0. The method works for certain subgroups of infinite index as well and we show $scl$ is uniformly positive on the nontrivial elements of the Torelli group. The proofs use our earlier construction in the paper Constructing group actions on quasi-trees and applications to mapping class groups of group actions on quasi-trees.
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We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.
We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from omega-signatures, and show that they are linearly independent in the mapping class groups of pointed 2-spheres when the number of points is small.
This is an addendum to arXiv: 0810.5376. We show, using our methods and an auxiliary result of Bestvina-Bromberg-Fujiwara, that a finitely generated group with infinitely many pairwise non-conjugate homomorphisms to a mapping class group virtually acts non-trivially on an $R$-tree, and, if it is finitely presented, it virtually acts non-trivially on a simplicial tree
This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^sharp,T^ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjians and Degenhardts conjectures by showing that $T^sharp,T^ast$ are of type $F_infty$ and that $mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.
We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.
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