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The Virtual Haken Conjecture: experiments and examples
 GEOM. TOPOL
, 2002
"... 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 ..."
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Cited by 44 (2 self)
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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.
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 5 (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...
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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this paper we will study the Smith Normal Form of the matrices B.
Computing nilpotent quotients in finitely presented Lie rings
 DISCRETE MATH. THEOR. COMPUT. SCI
, 1997
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Application of computational tools for finitely presented groups
 In Finkelstein and Kantor [Finkelstein and Kantor
, 1993
"... Abstract. Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined. They are available both in standalone programs and in more comprehensive systems. A general computational approach for investigating finitely presented gr ..."
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Abstract. Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined. They are available both in standalone programs and in more comprehensive systems. A general computational approach for investigating finitely presented groups by way of quotients and subgroups is described and examples are presented. The techniques can provide detailed information about group structure. Under suitable circumstances a finitely presented group can be shown to be soluble and its complete derived series can be determined, using what is in effect a soluble quotient algorithm. 1.
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 ..."
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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.
Efficient Parallelizations of Hermite and Smith Normal Form Algorithms
"... Hermite and Smith normal form are important forms of matrices used in linear algebra. These terms have many applications in group theory and number theory. As the entries of the matrix and of its corresponding transformation matrices can explode during the computation, it is a very difficult proble ..."
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Hermite and Smith normal form are important forms of matrices used in linear algebra. These terms have many applications in group theory and number theory. As the entries of the matrix and of its corresponding transformation matrices can explode during the computation, it is a very difficult problem to compute the Hermite and Smith normal form of large dense matrices. The main problems of the computation are the large execution times and the memory requirements which might exceed the memory of one processor. To avoid these problems, we develop parallelizations of Hermite and Smith normal form algorithms. These are the first parallelizations of algorithms for computing the normal forms with corresponding transformation matrices, both over the rings Z and F[x]. We show that our parallel versions have good efficiency, i.e., by doubling the processes, the execution time is nearly halved. Furthermore, they succeed in computing normal forms of dense large example matrices over the rings Q[x], F3[x], and F5[x].
computational complexity FAST COMPUTATION OF THE SMITH FORM OF A SPARSE INTEGER MATRIX
"... Abstract. We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈ Z m×n. The algorithm treats A as a “black box”—A is only used to compute matrixvector products and we do not access individual entries in A directly. The algorithm requires about O(m2 ..."
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Abstract. We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈ Z m×n. The algorithm treats A as a “black box”—A is only used to compute matrixvector products and we do not access individual entries in A directly. The algorithm requires about O(m2 log �A�) black box eval, plus uations w ↦ → Aw mod p for wordsized primes p and w ∈ Zn×1 p O(m2n log �A � + m3 log 2 �A�) additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. The new algorithm suffers from no “fillin ” or intermediate value explosion, and uses very little additional space. We also present an asymptotically fast algorithm for dense matrices which requires about O(n · MM(m) log �A � + m3 log2 �A�) bit operations, where O(MM(m)) operations are sufficient to multiply two m × m matrices over a field. Both algorithms are probabilistic of the Monte Carlo type—on any input they return the correct answer with a controllable, exponentially small probability of error.