Marc Culler

Dehn Surgery on Knots

In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that

Moduli of graphs and automorphisms of free groups

This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study mapping classes of surfaces. Let F, denote the free group of rank n. We will study the g roup Out(F,) of outer au tomorphisms of F, by studying its act ion on a space X, which is analogous to the Teichmtiller space of hyperbol ic metrics on a surface; the points of X, are metric structures on graphs with fundamental group F,. We begin by making this not ion precise. By a graph we shall mean a connected 1-dimensional CW-complex. The 0cells will be called nodes and the l-cells edges. The valence of a node x is the number of oriented edges which terminate at x, i.e. the min imum number of components of an arbitrarily small deleted ne ighborhood of x. An N-graph is a graph endowed with a metric such that each edge is locally isometric to an interval in l l and such that the distance between two points is the length of the shortest edge-path joining them. An N-graph is said to be minimal if it is not homotopy equivalent to any proper subgraph. Th roughou t this paper we will consider only ~,-graphs which are minimal and have no nodes of valence 2. (A minimal N-g raph cannot have nodes of valence 1). Fix a (topological) graph R o with one node and n edges, and choose an identification F--~rl(Ro). If G is an N-graph, then a homotopy equivalence g : R o ~ G is called a marking on G. We define two markings g l :Ro,G1 and g 2 : R o ~ G 2 to be equivalent if there exists an isometry i: GI~G 2 making the following diagram commute up to (free) homotopy :

Lifting Representations to Covering Groups

The primary purpose of this note is to prove that if a discrete subgroup Z of Z’S&(C) has no 2-torsion then it lifts to S&(C); i.e., there is a homomorphism Z+ S&(C) such that the composition with the natural projection S&(C) + Z’S&(C) is the identity on K (Note that the absence of 2-torsion is necessary, since cyclic subgroups of even order do not lift.) This extends a result of Km [2], for which Z was required to be a function group. Our methods are simple and straightforward, requiring only elementary covering space theory.

Using surfaces to solve equations in free groups

BECAUSE THERE exist groups like those described in [l], it is futile to attempt to study the solution of equations in an arbitrary group. It is reasonable, however, to do so for free groups. Results along these lines have been obtained by Edmonds, Lyndon, Malcev, Schupp, Wicks and others. With few exceptions the equations which have been successfully handled are equivalent to equations involving products of commutators or squares. The defining relations for fundamental groups of 2-manifolds have this form, which suggests that one should be able to use the theory of surfaces to study these equations. This is what will be done here. If g is an element of the commutator subgroup [G,G] of a G, we define Genus (g) to be the least integer n such that there exist elements al, bl, . . . ,a,, b, of G with g = [a,, b,]. . .[a,, b,]. (Here [a, b] denotes abu-‘b-l). Similarly, if g can be written as a product of squares, we define Sq (g) to be the least integer n such that there exist elements al,. . . ,a, of G with g = u12.. .u,,~. We begin by describing a homotopy classification of maps from a bounded surface to a l-complex when the surface satisfies certain minimality conditions. This is applied to give short proofs that there exist effective procedures for computing Genus (g) and Sq (g) when g is an element of a free group. The existence of such procedures was first shown by Edmunds [2,3] using cancellation arguments. These algorithms are used in some non-trivial (and somewhat surprising) examples. We then give an exact description of a set of standard forms for words of a given genus. The sets of all solutions to the equations g = [(Y,, p,]. . .[a”, pn] and g = (Y,~. an2 are described under the conditions Genus (g) = n and Sq (g) = n respectively. Finally, under suitable conditions on groups U and V, we show that there are effective procedures for computing Genus (g) and Sq (g) when g is an element of the free product U* V.

Bounded, separating, incompressible surfaces in knot manifolds

This generalizes and strengthens the main theorem of [13]. Note that the hypothesis of Theorem 1 is satisfied whenever M is a knot manifold, i.e. the complement of an open tubular neighborhood of a nontrivial knot in S 3. (The theorem of [13] gives no information for a knot manifold.) In this case various versions of Theorem 1 have been conjectured by L.P. Neuwirth. For example, Conjecture A of [11], that every knot group is a free product of two proper subgroups amalgamated along a free group, is an immediate corollary to Theorem 1. The following result can be derived from Theorem 1 above in the same way that [13, Theorem 2] is derived from [13, Theorem 1]. Because the proof parallels so precisely that of [13, Theorem 2], we shall omit it.

Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds

The 2-thin part of a hyperbolic manifold, for an arbitrary positive number 2, is defined to consist of all points through which there pass homotopically non-trivial curves of length at most 2. For small enough 2, the 2-thin part is geometrically very simple: it is a disjoint union of standard neighborhoods of closed geodesics and cusps. (Explicit descriptions of these standard neighborhoods are given in Section 1.) If 2 is small enough so that the 2-thin part of M has this structure then 2 is called a Margulis number of M. There is a positive number, called a 3-dimensional Margulis constant, which serves as a Margulis number for every hyperbolic 3-manifold. The results of this paper provide surprisingly large Margulis numbers for a wide class of hyperbolic 3-manifolds. In particular we obtain the following result, which is stated as Theorem 10.3:

A group theoretic criterion for property FA

We give group-theoretic conditions on a set of generators of a group G which imply that G admits no non-trivial action on a tree. The criterion applies to several interesting classes of groups, including automorphism groups of most free groups and mapping class groups of most surfaces.

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.

Dehn surgery, homology and hyperbolic volume

If a closed, orientable hyperbolic 3‐manifold M has volume at most 1.22 then H1.MIZp/ has dimension at most 2 for every prime p⁄ 2;7, and H1.MIZ2/ and H1.MIZ7/ have dimension at most 3. The proof combines several deep results about hyperbolic 3‐manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic C M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C . 57M50; 57M27

Volumes of hyperbolic Haken manifolds, I

In [14] a program was initiated for using the topological theory of 3-manifolds to obtain lower bounds for volumes of hyperbolic 3-manifolds. In [1], by a combination of new geometric ideas with relatively standard (but specifically 3-dimensional) topological techniques, we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is at least 3 has volume exceeding 0.92. By contrast, the best known lower bound [10,5] for the volume of an arbitrary closed hyperbolic 3-manifold is approximately 0.0012. In [3] we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is 2 has volume exceeding 0.34. The proof depended on supplementing the results and techniques of [1] with ingenious elementary arguments due to Zagier [11] and numerical computations. In the present paper we shall show that if one excludes certain special manifolds, such as fiber bundles over S 1, then the lower bound of 0.34 also holds for hyperbolic 3-manifolds with Betti number 1. The proof depends heavily on the results of [1] and [3], but it involves much deeper topological ideas than these papers. The new topological results needed for the proof occupy most of the present paper. To some extent these results have the flavor of general topology, but the proofs make use of such specifically low-dimensional techniques as the characteristic submanifold theory [9, 8], the interaction between trees and incompressible surfaces, and Scotts theorem [12] that surface groups are locally extended residually finite. Before giving a precise statement of our main result we must review a few elementary notions from 3-manifold theory.