We prove the Bers density conjecture for singly degenerate Kleinian surface groups without parabolics.

Let G be a finite index subgroup of the mapping class group M CG(Σ) of a closed orientable surface Σ, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element g ∈ G 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 [BBFb] of group actions on quasi-trees.1 of course, scl(g) = 0 if g k is conjugate to g l for some k = l, but in mapping class groups this is possible only when k = ±l or g has finite order

No abstract

We give a bound, linear in the complexity of the surface, to the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.

No abstract