Back to WTC Winter 2008

Ian Agol
Bounds on exceptional Dehn filling
Thurston showed that the number of non-hyperbolic (exceptional) Dehn fillings on the figure 8 knot is 10. It is conjectured that this is the only example of a one cusped hyperbolic manifold with 10 exceptional Dehn fillings. Recently Lackenby and Meyerhoff have shown that 10 is the maximal number of exceptional Dehn fillings. We show that there are only finitely many one cusped hyperbolic 3-manifolds with more than 8 exceptional Dehn fillings. Moreover, we show that there is an algorithm which would find all of these exceptions and therefore resolve the conjecture.

Shinpei Baba
Complex projective structures with Schottky holonomy
A Schottky group in PSL(2, C) induces an open hyperbolic handlebody and its ideal boundary is a closed orientable surface S whose genus is equal to the rank of the Schottky group. This boundary surface is equipped with a (complex) projective structure and its holonomy representation is an epimorphism from \pi_1(S) to the Schottky group. We will show that an arbitrary projective structure with the same holonomy representation is obtained by (2\pi-)grafting the basic structure described above.

This result is an analog to the characterization of the projective structures whose  holonomy representation is an isomorphism from \pi_1(S) to a fixed quasifuchsian group, which was given by Goldman in 1987.

Tom Church
Groups of mapping classes that cannot be realized by diffeomorphisms
Morita proved that the mapping class group cannot be realized by diffeomorphisms. The mapping class group of a surface S with one marked point z fits into the short exact sequence 1 --> pi_1(S,z) --> Map(S,z) --> Map(S) --> 1.The kernel is known as the point-pushing subgroup, since its elements are obtained by "pushing" the marked point along loops in the fundamental group of S. By using Milnor's inequality for the Euler number of a flat vector bundle over a surface, we show that the point-pushing subgroup cannot be realized by diffeomorphisms of S fixing z. We apply this result to construct a group isomorphic to pi_1(S') x Z/3Z inside Map(S) that cannot be realized by diffeomorphisms; as a corollary, this yields a new proof of Morita's theorem. Joint work with Mladen Bestvina and Juan Souto.

Matt Clay
Twisting out fully irreducible automorphisms
By a well-known theorem of Thurston, in the subgroup of the mapping class group generated by two Dehn twists about curves which fill the surface every element not conjugate to a power of one of the twists is pseudo-Anosov.  We prove a generalization of this theorem for the outer automorphism group of a free group.  This is joint work with Alexandra Pettet.

Alexi Eskin
Counting closed geodesics in strata
AWe obtain asymptotic formulas for the number of closed geodesics in a stratum of Teichmuller space. In particular, we compute the asymptotics, as R tends to infinity, of the number of pseudo-anosov elements of the mapping class group which have translation length less then R and a orientable invariant foliations. This is joint work in progress with Maryam Mirzakhani and Kasra Rafi.

Ilya Kapovich
Schottki-type subgroups of Out(F_n)
For outer automorphisms of a free group $F_N$ the best analog of being pseudo-ansov is being an atoriodal iwip. Here "atoroidal" means that the automorphism is without periodic conjugacy classes and "iwip" means being irreducible with all  nonzero powers being irreducible as well. By a result of Brinkmann for an element $\phi$ of $Out(F_N)$ being atoroidal is equivalent to being "hyperbolic" in the sense of the Bestvina-Feighn Combination Theorem which in turn is equivalent to the mapping toris of $\phi$ being word-hyperbolic. For elements of mapping class groups of closed surfaces being "hyperbolic" is the same thing as being irreducible of infinite order (that is, being pseudo-anosov). However, for free groups that is no longer the case and there are lots of reducible hyperbolic automorphisms.

We show that for an arbitrary finite collection of iwip atoroidal automorphisms $\phi_1,\dots \phi_k$ in $Out(F_N)$ with distinct "axes", sufficiently high powers of them generate a free subgroup of rank $k$ in Out(F_N) which is "purely atoriodal iwip", that is every nontrivial element of that subgroup is an atoroidal iwip. The "purely atoroidal" conclusion is already known by the result of Bestvina-Feighn-Handel, and we establish the "purely iwip" property. Time permitting, we will also discuss constructions of discontinuity domains in the boundary of the Outer space for arbitrary sufficiently large subgroups of $Out(F_N)$. This is joint work with Martin Lustig.