Abstract. In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph Γ, we produce a new graph through a purely combinatorial procedure, and call it the extension graph Γ e of Γ. We produce a second graph Γ e k , the clique graph of Γ e , by adding extra vertices for each complete subgraph of Γ e . We prove that each finite induced subgraph Λ of Γ e gives rise to an inclusion A(Λ) → A(Γ). Conversely, we show that if there is an inclusion A(Λ) → A(Γ) then Λ is an induced subgraph of Γ e k . These results have a number of corollaries. Let P 4 denote the path on four vertices and let C n denote the cycle of length n. We prove that A(P 4 ) embeds in A(Γ) if and only if P 4 is an induced subgraph of Γ. We prove that if F is any finite forest then A(F ) embeds in A(P 4 ). We recover the first author's result on co-contraction of graphs, and prove that if Γ has no triangles and A(Γ) contains a copy of A(C n ) for some n ≥ 5, then Γ contains a copy of C m for some 5 ≤ m ≤ n. We also recover Kambites' Theorem, which asserts that if A(C 4 ) embeds in A(Γ) then Γ contains an induced square. Finally, we determine precisely when there is an inclusion A(C m ) → A(C n ) and show that there is no "universal" two-dimensional right-angled Artin group.

Consider the mapping class group Modg,p of a surface Σg,p of genus g with p punctures, and a finite collection {f1, . . . , f k } of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N , the mapping classes {f N 1 , . . . , f N k } generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f1, . . . , f k } are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Modg,p, and prove that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Modg,p.2010 Mathematics Subject Classification. Primary 37E30; Secondary 20F36, 05C60.

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we are able to develop a Nielsen-Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of rightangled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.

Abstract. We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder condition for a finitely generated subgroup of A(Γ) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Γ. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups.

We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by C 1`bv diffeomorphisms on the circle, which generalizes a result of Farb-Franks, and which parallels a result of Ghys and Burger-Monod concerning differentiable actions of higher rank lattices on the circle. This answers a question of Farb, which has its roots in the work of Nielsen. We prove this result by showing that if a right-angled Artin group acts faithfully by C 1`bv diffeomorphisms on a compact one-manifold, then its defining graph has no subpath of length three. As a corollary, we also show that no finite index subgroup of AutpF n q and OutpF n q for n ě 3, the Torelli group for genus at least 3, and of each term of the Johnson filtration for genus at least 5, can act faithfully by C 1`bv diffeomorphisms on a compact one-manifold.