We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasiisometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.

Forester has defined spaces of simplicial tree actions for a finitely generated group, called deformation spaces. Culler and Vogtmann's Outer space is an example of a deformation space. Using ideas from Skora's proof of the contractibility of Outer space, we show that under some mild hypotheses deformation spaces are contractible. AMS Classification 20E08; 20F65, 20F28Keywords G-tree, deformation space, Outer space Culler and Vogtmann's Outer space is a good geometric model for Out(F n ), the outer automorphism group of a finitely generated free group of rank n ≥ 2, for three reasons:(1) Outer space is contractible;(2) point stabilizers are finite; and (3) there is a equivariant deformation retract on which the action is cocompact [4].

Abstract. We prove that the curve graph C(1) (S) is Gromovhyperbolic with a constant of hyperbolicity independent of the surface S. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.

For a fully irreducible automorphism φ of the free group F k we compute the asymptotics of the intersection number n → i(T, T φ n ) for the trees T and T in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T φ n for n large.

We generalize the familiar notion of a Whitehead move from Culler and Vogtmann's Outer space to the setting of deformation spaces of G‐trees. Specifically, we show that there are two moves, each of which transforms a reduced G‐tree into another reduced G‐tree, that suffice to relate any two reduced trees in the same deformation space. These two moves further factor into three moves between reduced trees that have simple descriptions in terms of graphs of groups. This result has several applications.