The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the continuum hypothesis. This answers a longstanding problem of Shelah.

We consider$Thurston maps$, i.e., branched covering maps$f:S$^{2}→$S$^{2}that are$post-critically finite$. In addition, we assume that$f$is$expanding$in a suitable sense. It is shown that each sufficiently high iterate$F$=$f$^{$n$}of$f$is$semi-conjugate$to$z$^{$d$}:$S$^{1}→$S$^{1}, where$d$= deg F. More precisely, for such an$F$we construct a$Peano curve γ$:$S$^{1}→$S$^{2}(onto), such that$F$∘$γ$($z$) =$γ$($z$^{$d$}) (for all$z$∈$S$^{1}).

Let$k$be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane $$ \mathbb{P}_{\mathbf{k}}^2 $$ is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π_{2}-sentences over the structure ($H$($ω$_{2}), ∈, NS_{$ω$1}), in the sense that its ($H$($ω$_{2}), ∈, NS_{$ω$1}) satisfies every Π_{2}-sentence$σ$for which ($H$($ω$_{2}), ∈, NS_{$ω$1}) ⊨$σ$can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π_{2}-sentences over the structure ($H$($ω$_{2}), ∈,$ω$_{1}) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $$ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $$ . In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.

We give a complete proof of the Bers–Sullivan–Thurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence. We derive the necessary compactness theorem combining the Rips machine with non-existence results for certain small actions on real trees of free products of surface groups and free groups. We prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface. In order to be able to apply the ending lamination theorem, we have to prove that this algebraic limit has the desired topological type and that these non-realized laminations are ending laminations. That this is the case is the main novel technical result of this paper. Loosely speaking, we show that a filling Masur domain lamination which is not realized is an ending lamination.