Thurston equivalence of topological polynomials

We answer Hubbard's question on determining the Thurston equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with nth powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.     

On Dehn Functions of Infinite Presentations of Groups

We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, for two well-known periodic groups. In particular, we prove that the (standard) Dehn function of a 2-group Γ of intermediate growth, defined by a system of defining relators due to Lysenok, is bounded from above by C₁x² log₂ x, where C₁ > 1 is a constant. We also show that the (standard) Dehn function of a free m-generator Burnside group B(m, n) of exponent n ≥ 2⁴⁸, where n is either odd or divisible by 2⁹, defined by a minimal system of defining relators, is bounded from above by the subquadratic function x^19/12. Received: September 2007, Revision: March 2008, Accepted: March 2008     

Groups with small Dehn functions and bipartite chord diagrams

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n²log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n²log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem. Both authors were supported in part by the NSF grants DMS 0245600 and DMS 0455881. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 05-01-00895, the research of the second author was supported in part by the NSF grant DMS 9978802 and the US-Israeli BSF grant 1999298. Received: February 2005; Revision: September 2005; Accepted: September 2005     

Closed minimal surfaces in cusped hyperbolic three-manifolds

We introduce essential open book foliations by refining open book foliations, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the ‘gap’ between overtwisted contact structures and non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3-manifold and the complement of a closed braid are determined by the Nielsen–Thurston types of the monodromies of their open book decompositions.     

Peripheral fillings of relatively hyperbolic groups

In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π₁(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Mathematics Subject Classification (2000) 20F65, 20F67, 20F06, 57M27, 20E26     

The Dehn Function of Stallings’ Group

We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F₃{\mathcal{F}_3} is quadratic.     

Twisting Out Fully Irreducible Automorphisms

By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.     

Dehn twists and pseudo-Anosov diffeomorphisms

Summary We show that it is possible to obtain many pseudo-Anosov diffeomorphisms from Dehn twists. In particular, we generalize a theorem of Long and Morton to obtain that iff is a pseudo-Anosov diffeomorphism of an oriented surface andT is the Dehn twist around the simple closed curve , then the isotopy class ofT ⁿ f contains a pseudo-Anosov diffeomorphism except for at most 7 consecutive values ofn.     

Dehn surgeries that yield fibred 3-manifolds

We study Dehn surgeries on null-homotopic knots that yield fibred 3-manifolds when an additional (but natural) homological restriction is imposed. The major tool used is Gabai’s theory of sutured manifold decomposition. Such surgeries are negative examples to a question of Michel Boileau. Another result we will prove is about surgeries which reduce the Thurston norm of a fibred manifold.     

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of . Received: February 12, 1996.     

Rectangling a rectangle

We show that the following are equivalent: (i) A rectangle of eccentricityv can be tiled using rectangles of eccentricityu. (ii) There is a rational function with rational coefficients,Q(z), such thatv =Q(u) andQ maps each of the half-planes {z | Re(z) < 0} and {z | Re(z) > 0 into itself, (iii) There is an odd rational function with rational coefficients,Q(z), such thatv = Q(u) and all roots ofv = Q(z) have a positive real part. All rectangles in this article have sides parallel to the coordinate axes and all tilings are finite. We letR(x, y) denote a rectangle with basex and heighty. In 1903 Dehn [1 ] proved his famous result thatR(x, y) can be tiled by squares if and only ify/x is a rational number. Dehn actually proved the following result. (See [4] for a generalization to tilings using triangles.) The first and third authors were partially supported by NSF.     

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.     

Linear Equations for the Number of Intervals Which are Isomorphic with Boolean Lattices and the Dehn–Sommerville Equations

Let P be a finite poset. Let L: = J(P) denote the lattice of order ideals of P. Let b _i (L) denote the number of Boolean intervals of L of rank i.

We construct a simple graph G(P) from our poset P. Denote by f _i (P) the number of the cliques K _i+1, contained in the graph G(P).

Our main results are some linear equations connecting the numbers f _i (P) and b _i (L).

We reprove the Dehn–Sommerville equations for simplicial polytopes.

In our proof, we use free resolutions and the theory of Stanley–Reisner rings.     

Complexity of Geometric Three-manifolds

We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev's complexity implied by our computations are sharp for thousands of manifolds, and we conjecture they are for infinitely many, including all Seifert manifolds. Our computations and estimates apply to all the Dehn fillings of M 6 ₁ ³ (the complement of the three-component chain-link, conjectured to be the smallest triply cusped hyperbolic manifold), whence to infinitely many among the smallest closed hyperbolic manifolds. Our computations are based on the machinery of the decomposition into ‘bricks’ of irreducible manifolds, developed in a previous paper. As an application of our results we completely describe the geometry of all 3-manifolds of complexity up to 9.     

Topological imitation of a colored link with the same Dehn surgery manifold

By the topological imitation theory, we construct, from a given colored link, a new colored link with the same Dehn surgery manifold. In particular, we construct a link with a distinguished coloring whose Dehn surgery manifold is a given closed connected oriented 3-manifold except the 3-sphere. As a result, we can naturally generalize the difference between the Gordon–Luecke theorem and the property P conjecture to a difference between a link version of the Gordon–Luecke theorem and the Poincaré conjecture. Similarly, we construct a link with a π₁-distinguished coloring whose Dehn surgery manifold is a given non-simply-connected closed connected oriented 3-manifold. We also construct a link with just two colorings whose Dehn surgery manifolds are the 3-sphere.     

Reidemeister torsion for linear representations and Seifert surgery on knots

We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over S². As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information.     

The classification of Dehn fillings on the outer torus of a 1-bridge braid exterior which produce solid tori

Let K=K(w,b,t) be a 1-bridge braid in a solid torus V, and let be a (p,q) curve on the torus T=V of the exterior M_K of K. It will be shown that Dehn filling on T along produces a solid torus if and only if p and q satisfy one of four conditions determined by the parameters (w,b,t) of the knot K. This solves the classification problem raised by Menasco and Zhang for such Dehn fillings.Mathematics Subject Classification (1991): 57N10Partially supported by NSF grant DMS 0203394Revised version: 28 November 2003     

On Quasiperiodic Space Tilings, Inflation, and Dehn Invariants

We introduce Dehn invariants as a useful tool in the study of the inflation of quasiperiodic space tilings. Tilings by ``golden tetrahedra' are considered. We discuss how Dehn invariants can be applied to the study of the inflation properties of the six golden tetrahedra. We also use the geometry of the faces of the golden tetrahedra to analyze their inflation properties. We give inflation rules for decorated Mosseri—Sadoc tiles in the projection class of tilings <I>T<SUP>(MS)</SUP></I> (the tiles of <I>T<SUP>(MS)</SUP></I> are particular packages of the golden tetrahedra). The Dehn invariants of the Mosseri—Sadoc tiles provide two eigenvectors of the inflation matrix with eigenvalues equal to \t = (1+\sqrt 5 )/2 and -1/\t , and allow us to reconstruct the inflation matrix uniquely. Received October 11, 1999, and in revised form July 5, 2000. Online publication May 4, 2001.     

Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori. S. Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00024). M. Teragaito was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19540089.