Ian Agol

Criteria for virtual fibering

We prove that an irreducible 3-manifold with fundamental group that satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber dodecahedral space and the Bianchi groups. Moreover, we show that a taut-sutured compression body has a finite-sheeted cover with a depth one taut-oriented foliation.

Bounds on exceptional Dehn filling II

We show that for a hyperbolic knot complement, all but at most 12 Dehn llings are irreducible with innite word-hyperbolic fundamental group.

The minimal volume orientable hyperbolic 2-cusped 3-manifolds

We prove that the Whitehead link complement and the pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume = 4 Catalans constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.

Singular surfaces, mod 2 homology, and hyperbolic volume, I

If M is a simple, closed, orientable 3-manifold such that π 1 (M) contains a genus-g surface group, and if H 1 (M; ℤ 2 ) has rank at least 4g—1, we show that M contains an embedded closed incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the rank of H 1 (M; ℤ 2 ) is at most 6.

Presentation length and Simons conjecture

In this paper, we show that any knot group maps onto at most finitely many knot groups. This gives an affirmative answer to a conjecture of J. Simon. We also bound the diameter of a closed hyperbolic 3-manifold linearly in terms of the presentation length of its fundamental group, improving a result of White.

Residual finiteness, QCERF and fillings of hyperbolic groups

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling. 20E26, 20F67, 20F65 A group G is residually finite (or RF) if for every g2 GXf1g, there is some finite group F and an epimorphism W G! F so that . g/⁄ 1. In more sophisticated language G is RF if and only if the trivial subgroup is closed in the profinite topology on G . If H < G , then H is separable if for every g2 GXH , there is some finite group F and an epimorphism W G! F so that . g/O. H/. Equivalently, the subgroup H is separable in G if it is closed in the profinite topology on G . If every finitely generated subgroup of G is separable, G is said to be LERF or subgroup separable. If G is hyperbolic, and every quasi-convex subgroup of G is separable, we say that G is QCERF. In this paper, we show that if every hyperbolic group is RF, then every hyperbolic group is QCERF.

Finiteness of arithmetic hyperbolic reflection groups

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

Pseudo-Anosov stretch factors and homology of mapping tori

We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k+1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who treated the cases of k=0 and k=2g, respectively, and answers a question of Ellenberg. We also show that the number of conjugacy classes of pseudo-Anosov mapping classes as above grows (as a function of g) like a polynomial of degree k.