Richard P. Kent

A GEOMETRIC AND ALGEBRAIC DESCRIPTION OF ANNULAR BRAID GROUPS

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph An - 1. This provides a new example of an infinite type Artin group which injects into a finite type Artin group. In fact, we show that the affine braid group with Coxeter graph An - 1 injects into the braid group on n + 1 stings. Recently it has been shown that the braid groups are linear, see [3]. Therefore, this shows that the affine braid groups are also linear.

Trees and mapping class groups

Abstract There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of B. Farb and L. Mosher. In particular, we obtain an affirmative answer to their question of local convex cocompactness of K. Whittleseys group. In the course of the proof, we obtain a new proof of a theorem of I. Kra. We also relate the action of this kernel on the curve complex to a family of actions on trees. This quickly yields a new proof of a theorem of J. Harer.

Pseudo-Anosov subgroups of fibered 3-manifold groups

Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing a point from X. We prove that a subgroup G in H is convex cocompact in Mod(Y) if and only if G is finitely generated and purely pseudo-Anosov. We also prove a generalization of this theorem with H replaced by an arbitrary Gromov hyperbolic extension of the fundamental group of X, and an additional hypothesis of quasi-convexity of G in H. Along the way, we obtain a generalization of a theorem of Scott and Swarup on the geometric finiteness of subgroups of fibered 3-manifold groups.

Uniform convergence in the mapping class group

We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.

Intersections and joins of free groups

Let F be a free group. If H and K are subgroups of F , we let H ∨K = 〈H,K〉 denote the join of H and K. We study the relationship between the rank of H ∩K and that of H ∨K for a pair of finitely generated subgroups H and K of F . In particular, we have the following particular case of the Hanna Neumann Conjecture, which has also been obtained by L. Louder [6] using his machinery for folding graphs of spaces [7, 8, 9]. For detailed discussions of the Hanna Neumann Conjecture, see [11, 12, 13, 16, 4, 3]. Theorem 1 (Kent, Louder). Let H and K be nontrivial finitely generated subgroups of a free group of ranks h and k, respectively. If rank(H ∨K)−1≥ h+ k−1 2 then

Surface groups are frequently faithful

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL2(K) representation variety, where K = C or R, answering a question of W. Goldman. We also prove the existence of faithful representations into PU(2, 1) with certain nonintegral Toledo invariants.

Bers slices are Zariski dense

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

Totally geodesic boundaries of knot complements

Given a compact orientable 3 manifold M whose boundary is a hyperbolic surface and a simple closed curve C in its boundary, every knot in M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which the geodesic representative of C is as small as you like.

Bundles, Handcuffs, and Local Freedom

We answer a question of J. Andersons by producing infinitely many commensurability classes of fibered hyperbolic 3-manifolds whose fundamental groups contain subgroups that are locally free and not free. These manifolds are obtained by performing 0–surgery on a collection of knots with the same properties.

ACHIEVABLE RANKS OF INTERSECTIONS OF FINITELY GENERATED FREE GROUPS

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups. Let F be a free group. Let H and K be nontrivial finitely generated subgroups of F . It is a theorem of Howson [1] that H ∩ K has finite rank. H. Neumann proved in [2] that rank(H ∩ K) − 1 ≤ 2(rank(H) − 1)(rank(K) − 1) and asked whether or not rank(H ∩ K) − 1 ≤ (rank(H) − 1)(rank(K) − 1). A. Myasnikov has asked which values between 1 and (m − 1)(n − 1) can be achieved as rank(H ∩ K) − 1 for subgroups H and K of ranks m and n—this is problem AUX1 of [4]. We prove that all such numbers occur by proving the following Theorem. Let F (a, b) be a free group of rank two. Let H k,` = 〈a, bab , . . . , bab, bab, bab, bab, . . . , bab〉 and let K = 〈b, aba, . . . , aba〉, where 0 ≤ k ≤ m − 2 and 0 ≤ ` ≤ n − 1. Then the rank of H k,` ∩ K is k(n − 1) + `. Corollary. Let F be a free group and let m, n ≥ 2 be natural numbers. Let N be a natural number such that 1 ≤ N − 1 ≤ (m − 1)(n − 1). Then there exist subgroups H, K ≤ F , of ranks m and n, such that the rank of H ∩ K is N . Proof of the corollary. The theorem produces the desired subgroups for all N with N − 1 ≤ (m− 1)(n− 1)− 1 after passing to a rank two subgroup of F . For N − 1 = (m− 1)(n− 1), simply let H = 〈a, bab, . . . , bab, b〉 and let K = 〈b, aba, . . . , aba, a〉. Proof of the theorem. Let X be a wedge of two circles and base π1(X) at the wedge point. We identify π1(X) with F = F (a, b) by calling the homotopy classWe answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups.