Examples of non-trivial roots of unity at ideal points of hyperbolic 3-manifolds

This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points whose associated roots of unity are not 1 or -1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than 1 and -1 occur.Comment: 12 pages, 1 figure, LaTeX2e. Minor changes, additional remarks, new description of 2nd example. To appear in_Topology

Laminations and groups of homeomorphisms of the circle

If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that pi_1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so pi_1(M) is isomorphic to a subgroup of Homeo(S^1). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S^1. As a corollary, the Weeks manifold does not admit a tight essential lamination, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston's universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem.Comment: 50 pages, 12 figures. Ver 2: minor improvement

Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration of M over the circle is the modular elliptic curve E=X_0(49), which admits multiplication by the ring of integers of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of level 7 of the multiplicative group of a maximal order of D. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a non-compact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.Comment: 42 pages, 7 figures; V2: minor improvements, to appear in Amer. J. Mat

An ascending HNN extension of a free group inside SL(2,C)

We give an example of a subgroup of SL(2,C) which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL(2,C) of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir. The main ingredient in our construction is a specific finite volume (noncompact) hyperbolic 3-manifold M which is a surface bundle over the circle. In particular, most of F comes from the fundamental group of a surface fiber. A key feature of M is that there is an element of its fundamental group with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group F we construct is actually free.Comment: 7 pages. V2: minor improvements in expositio