Peter B. Shalen

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

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.

Dendrology of Groups: An Introduction

The study of group actions on “generalized trees” or “ℝ-trees” has recently been attracting the attention of mathematicians in several different fields. This subject had its beginnings in the work of R. Lyndon [L] and I. Chiswell [Chi], and—from a different point of view—in the work of J. Tits [Ti]. The link between the points of view of these authors was provided by R. Alperin and K. Moss in [AM]. J. Morgan and I, in [MoS1,Mo2], established connections of this theory with hyperbolic geometry and with W. Thurston’s theory of measured laminations. The picture has been developed further by the above-mentioned people and also by H. Bass, M. Bestvina, M. Culler, H. Gillet, M. Gromov, W. Parry, F. Rimlinger and J. Stallings, among others.

Generalized triangle groups

A group G is called a triangle group if it can be presented in the form It is well-known that G is isomorphic to a subgroup of PSL 2 (ℂ), that a is of order l, b is of order m and ab is of order n . If then G contains the fundamental group of a positive genus orientable surface as a subgroup of finite index whenever s ( G ) ≤ 1; in particular G is infinite. Furthermore, if s ( G ) G contains a free group of rank 2.

Growth rates, _{}-homology, and volumes of hyperbolic 3-manifolds

It is shown that if M is a closed orientable irreducible 3-manifold and n is a nonnegative integer, and if H 1 (M, Z p ) has rank ≥ n+2 for some prime p, then every n-generator subgroup of π 1 (M) has infinite index in π 1 (M), and is in fact contained in infinitely many finite-index subgroups of π 1 (M). This result is used to estimate the growth rates of the fundamental group of a 3-manifold in terms of the rank of the Z p -homology

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:

Free actions of surface groups on R-trees

IN [9] Lyndon introduced a class of real-valued functions on groups, now called Lyndon length functions, and showed that a group is free if and only if it admits an integer-valued Lyndon length function. He raised the question of which groups admit R-valued Lyndon length functions. Using the construction of Chiswell [3] this question can be re-interpreted as asking which groups act freely (by isometries) on R-trees. (For the definition of an R-tree see [l2].) The examples of such groups pointed out by Lyndon arc arbitrary fret products of subgroups of R. The first examples not of this type wcrc pivcn by Alpcrin Moss in [Z]. Their examples arc not finitely gencratcd. In this paper WI: give the first finitely gcncratcd examples which are not free products of free abclian groups.

Linear groups of finite cohomological dimension

Our main result provides necessary and sufficient conditions for a finitelygenerated subgroup of GLn(C), n > 0, to have finite virtual cohomological dimension. A group has finite virtual cohomological dimension (VCD) if it has a subgroup of finite index which has finite cohomological dimension; this dimension is, in fact, the same for all torsion-free subgroups of finite index. It is, of course, necessary for a group r with VCD(T) < °° to have torsion-free subgroups of finite index; this is guaranteed in the case of finitely-generated linear groups by a well-known result of Selberg which extends ideas of Minkowski. A subgroup of GLn(C) is called unipotent if it is contained in a conjugate of the group of upper triangular matrices with all diagonal entries equal to one. Any unipotent subgroup is nilpotent; hence, a finitely-generated unipotent subgroup is poly cyclic and torsion-free. It is well known that a poly cyclic group has finite cohomological dimension if and only if it is torsion-free; moreover, the cohomological dimension is the same as the Hirsch rank. For a solvable group T with solvable series, 1 = Tn < Fn_l < • • • < I \ = T, the Hirsch rank, h(T) = Sĵ Tj dimQ(ryr /+1 <8> Q), is independent of the choice of solvable series; thus, for a polycyclic group r, h(T) is the number of infinite factors in a normal series with cyclic quotients. We announce our main result.

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