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Register a new userSurface groups, infinite generating sets, and stable commutator length
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We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.
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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.
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.
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.
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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