Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3-spheres. Specifically, for a compact 3-manifold $M$ with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of $M$ have left-orderable fundamental groups. Our technique uses certain representations from $\pi_1(M)$ into $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$, which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$ called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.

We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M < 3.44, then H_1(M;Z/2Z) has dimension at most 7.

It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it was the epoch of cohomology, it was the season of Ext, it was the season of Tor, it was the spring of short exact sequences, it was the winter of long exact sequences, we had right exactness, we had left exactness, our arrows were all going in one direction, our arrows were all reversed – in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, as the age of homological algebra.

For every closed orientable hyperbolic Haken 3-manifold and, more generally, for any orientable hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If H 1(M;ℚ) ≠ 0, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.

If g is an integer ⩾2, and M is a closed simple 3-manifold such that π1(M) has a subgroup isomorphic to a genus-g surface group and dimZ2H1(M;Z2)⩾max(3g−1,6), we show that M contains a closed, incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold such that VolM⩽3.08, then dimZ2H1(M;Z2)⩽5.

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that π_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. This result should be viewed as an analogue of Dehn's Lemma for π_1-injective singular surfaces. The geometric application states that if M is a closed orientable hyperbolic 3-manifold with volume less than 3.08 then the rank of H_1(M;Z/2Z) is at most 6. The proof of the geometric theorem combines the topological theorem with several deep geometric results, including the Marden tameness conjecture,recently established by Agol and by Calegari-Gabai; a co-volume estimate for 3-tame, 3-free Kleinian groups due to Anderson, Canary, Culler and Shalen; and a volume estimate for hyperbolic Haken manifolds recently proved by Agol, Storm and W. Thurston.

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes (i.e guts) of the complement of an incompressible surface in M.

We give lower bounds on the maximal injectivity radius for a closed hyperbolic 3-manifold with first Betti number 2 under some additional topological hypotheses.

Let M be a one-cusped hyperbolic 3-manifold. A slope on the boundary of the compact core of M is called exceptional if the corresponding Dehn filling produces a non-hyperbolic manifold. We give new upper bounds for the distance between two exceptional slopes alpha and beta in several situations. These include cases where M(beta) is reducible and where M(alpha) has finite pi(1), or M(alpha) is very small, or M(alpha) admits a pi(1)-injective immersed torus.

Let M be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that pi_1(M) has no subgroup isomorphic to a genus-2 surface group, and that either (a) H_1(M;Z_p) has dimension at least 5 for some prime p, or (b) H_1(M;Z_2) has dimension at least 4, and the subspace of H^2(M;Z_2) spanned by the image of the cup product has dimension at most 1, then vol M > 5.06 If we assume that H_1(M;Z_2) has dimension at least 7, and that the compact core of M does not contain a genus-2 closed incompressible surface, then vol M > 5.06.

At the very least, a geometric theory of manifolds would include a notion of distance, which could be expected to take the form of a metric that generates the topology of the manifold’s underlying topological space. In an interesting geometric theory this metric would be a path metric, meaning that there is a notion of the length of a path, and that the distance between two points is the infimum of the lengths of the paths joining them. In fact one would expect to have a notion of “straight lines” or geodesics, such that the distance between two points is realized by a path contained in a geodesic.

Many results in the theory of Dehn surgery (see [Go1]) assert that if M is a compact, orientable, atoroidal, irreducible 3-manifold whose boundary is an incompressible torus, and if two Dehn fillings M(α) and M(β) have specified properties, then the distance ∆(α, β) of the slopes α and β is bounded by a suitable constant. (The reader is referred to the body of this paper for the definition of “slope” and “distance”, as well as for precise versions of many definitions, statements and proofs that are hinted at in this introduction.)

Let M M be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in M M which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of M ( α ) M(\alpha ) has no non-abelian free subgroup, and if M ( β ) M(\beta ) is a reducible manifold which is not homeomorphic to S 1 × S 2 S^1 \times S^2 or P 3 # P 3 P^3 \# P^3 , then Δ ( α , β ) ≤ 5 \Delta (\alpha , \beta )\le 5 . Under the same condition on M ( β ) M(\beta ) , it is shown that if M ( α ) M(\alpha ) is Seifert fibered, then Δ ( α , β ) ≤ 6 \Delta (\alpha , \beta )\le 6 . Moreover, in the latter situation, character variety techniques are used to characterize the topological types of M ( α ) M(\alpha ) and M ( β ) M(\beta ) in case the bound of 6 6 is attained.

@We show that for a twist knot, the A-polynomial can be obtained from recurrences for the summand in Masbaum’s formula of the colored Jones polynomial. Our result supports the AJ conjecture due to S.Garoufalidis.

We consider irreducible 3-manifolds M that arise as knot complements in closed 3-manifolds and that contain at most two connected strict essential surfaces. The results in the paper relate the boundary slopes of the two surfaces to their genera and numbers of boundary components. Explicit quantitative relationships, with interesting asymptotic properties, are obtained in the case that M is a knot complement in a closed manifold with cyclic fundamental group.

Introduction. The group π1(S − k) of a knot k contains an extraordinary amount of information. From combined results of W. Whitten [Wh] and M. Culler, C. McA. Gordon, J. Luecke and P.B. Shalen [CuGoLuSh] it is known that there are at most two distinct unoriented prime knots with isomorphic groups. Unfortunately, knot groups are generally difficult to use. Knot groups are usually described by presentations, and there is no practical algorithm to decide whether or not two knot groups are isomorphic. In 1928 J.W. Alexander used homomorphisms (representations) of knot groups onto better understood groups in order to obtain topological invariants. Since then knot group representations have been used effectively by many others. The representations of a given knot group into a fixed finite group have the additional attraction that they are finite in number and so can be tabulated. R. Riley began such a program in [Ri]. We take a new approach, examining the representations of the commutator subgroup K = [π1(S − k), π1(S − k)] into a fixed finite group Σ. Although Hom(K, Σ) is often infinite – in fact, uncountable – it has a rich structure that we can understand via symbolic dynamics. In this dynamical system the representations of the knot group π1(S − k) appear (by restricting their domains) as special periodic points. However, the system contains other periodic points and often nonperiodic points, information that can be used to understand more about the structure of the knot exterior and its various covering spaces. The techniques, all algorithmic, apply equally well to links.

Let M be a compact, oriented, irreducible, atoroidal 3-manifold with nonempty boundary. Let CC0(M ) denote the space of convex cocompact Kleinian groups uniformizing M . We show that any Kleinian group in the boundary of CC0(M ) whose limit set is the whole sphere can be approximated by maximal cusps. Density of maximal cusps on the boundary of Schottky space is derived as a corollary. We further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3-manifold with connected conformal boundary.

The results in this paper show that simple connectivity of a 3-manifold is reflected in the behavior of essential surfaces in exteriors of knots in the manifold. A corollary of the main theorem is that any non-trivial knot, with irreducible complement, in a homotopy 3-sphere must have two boundary slopes that differ by at least 2. This statement is false for knots in a homology 3-sphere. The main theorem itself applies more generally to knots in closed orientable 3-manifolds with cyclic fundamental group.