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The Virtual Haken Conjecture: experiments and examples
 Geom. Topol
"... ABSTRACT. A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we desc ..."
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Cited by 34 (3 self)
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ABSTRACT. A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3manifolds. We took the complete HodgsonWeeks census of 10,986 smallvolume closed hyperbolic 3manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every nontrivial Dehn surgery on the figure8 knot is virtually Haken.
Cohomology of congruence subgroups of SL4(Z
 J. Number Theory
"... Abstract. In two previous papers we computed cohomology groups H5 (Γ0(N); C) for a range of levels N, where Γ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N. In this note we update this earlier work by carrying it out for prime lev ..."
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Cited by 10 (6 self)
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Abstract. In two previous papers we computed cohomology groups H5 (Γ0(N); C) for a range of levels N, where Γ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N. In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to H5 (Γ0(N); C) forNprime coming from Eisenstein series and Siegel modular forms. 1.
Groups of Deficiency Zero
 in Geometric and Computational Perspectives on Infinite Groups, DIMACS series in Discrete Mathematics and Theoretical Computer Science 25
, 1996
"... . We make a systematic study of groups of deficiency zero, concentrating on groups of primepower order. We prove that a number of pgroups have deficiency zero and give explicit balanced presentations for them. This significantly increases the number of such groups known. We describe a reasonably ..."
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Cited by 3 (1 self)
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. We make a systematic study of groups of deficiency zero, concentrating on groups of primepower order. We prove that a number of pgroups have deficiency zero and give explicit balanced presentations for them. This significantly increases the number of such groups known. We describe a reasonably general computational approach which leads to these results. We also list some other finite groups of deficiency zero. 1. Introduction In this paper we show how the use of symbolic computation changes the way in which one can attack previously intractable problems on group presentations. The group defined by a finite presentation fX : Rg is wellknown to be infinite if jX j ? jRj. A group is said to have deficiency zero if it has a finite presentation fX : Rg with jX j = jRj and jY j jSj for all other finite presentations fY : Sg of it. A presentation with the same number of generators and relators is called balanced. The generator number of a group G is the cardinality of a smallest gen...
Computing nilpotent quotients in finitely presented Lie rings
 DISCRETE MATH. THEOR. COMPUT. SCI
, 1997
"... ..."
On The Smith Normal Form Of The Varchenko Bilinear Form Of A Hyperplane Arrangement
, 1997
"... this paper we will study the Smith Normal Form of the matrices B. ..."
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Cited by 2 (1 self)
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this paper we will study the Smith Normal Form of the matrices B.
Article electronically published on January 20, 2010 COHOMOLOGY OF CONGRUENCE SUBGROUPS OF SL4(Z). III
"... Abstract. In two previous papers we computed cohomology groups H5 (Γ0(N); C) for a range of levels N, whereΓ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N. In this note we update this earlier work by carrying it out for prime leve ..."
Abstract
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Abstract. In two previous papers we computed cohomology groups H5 (Γ0(N); C) for a range of levels N, whereΓ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N. In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to H5 (Γ0(N); C) forNprime coming from Eisenstein series and Siegel modular forms. 1.
Abstract The virtual Haken conjecture: Experiments and examples
, 2003
"... A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe compu ..."
Abstract
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A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3manifolds. We took the complete HodgsonWeeks census of 10,986 smallvolume closed hyperbolic 3manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every nontrivial Dehn surgery on the figure8 knot is virtually Haken.