Matt Clay

Uniform hyperbolicity of the curve graph via surgery sequences

We prove that the curve graph C (1) (S) is Gromov- hyperbolic 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.

Growth of intersection numbers for free group automorphisms

For a fully irreducible automorphism φ of the free group Fk we compute the asymptotics of the intersection number n �→ i(T, Tφ 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.

On the isomorphism problem for generalized Baumslag-Solitar groups

Generalized Baumslag-Solitar groups (GBS groups) are groups that act on trees with infinite cyclic edge and vertex stabilizers. Such an action is described by a labeled graph (essentially, the quotient graph of groups). This paper addresses the problem of determining whether two given labeled graphs define isomorphic groups; this is the isomorphism problem for GBS groups. There are two main results and some applications. First, we find necessary and sufficient conditions for a GBS group to be represented by only finitely many reduced labeled graphs. These conditions can be checked effectively from any labeled graph. Then we show that the isomorphism problem is solvable for GBS groups whose labeled graphs have first Betti number at most one.

Whitehead moves for G-trees

We generalize the familiar notion of a Whitehead move from Culler and Vogtmanns 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.

Current twisting and nonsingular matrices

We show that for k at least 3, given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z^k is the given matrix.

Deformation spaces of G-trees and automorphisms of Baumslag–Solitar groups

We construct an invariant deformation retract of a deformation space of G-trees. We show that this complex is finite dimensional in certain cases and provide an example that is not finite dimensional. Using this complex we compute the automorphism group of the classical non-solvable Baumslag–Solitar groups BS.p; q/. The most interesting case is when p properly divides q. Collins and Levin computed a presentation for Aut.BS.p; q// in this case using algebraic methods. Our computation uses Bass–Serre theory to derive these presentations. Additionally, we provide a geometric argument showing Out.BS.p; q// is not finitely generated when p properly divides q. Mathematics Subject Classification (2000). 20E08,20F65,20F28.

RELATIVE TWISTING IN OUTER SPACE

Subsurface projection is indispensable to studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the thin part of Culler–Vogtmanns Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.

Stable commutator length in Baumslag–Solitar groups and quasimorphisms for tree actions

This paper has two parts, on Baumslag-Solitar groups and on general G-trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces. In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2,3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions. Returning to Baumslag-Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0, 1/12).

Uniform fellow traveling between surgery paths in the sphere graph

We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most 2. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most 4. Our proof relies on understanding how surgeries affect the Guirardel core associated to sphere systems. We show that applying a surgery is equivalent to performing a Rips move on the Guirardel core.