We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.

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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25

Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs. © 2001 Academic Press Key Words: formal proofs; logical flow graphs; cut elimination; bridge groups; Baumslag–Solitar groups; Gersten groups.

This book is based upon a set of lecture notes for a course that I was teaching at the University of Utah in Fall of 2002. Our main goal is to describe various tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental theorems in this area: Gromov’s theorem on groups of polynomial growth, Mostow Rigidity Theorem and Schwartz’s quasi-isometric rigidity theorem for nonuniform lattices in the real-hyperbolic spaces. We conclude with a survey of the quasi-isometric rigidity theory. The main idea of the geometric group theory is to treat finitely-generated groups as geometric objects: with each finitely-generated group G we associate a metric space, the Cayley graph of G. One of the main issues of the geometric group theory is to recover as much as possible algebraic information about G from the geometry of the Cayley graph. A primary obsticle for this is the fact that the Cayley graph depends not only on G but on a particular choice of a generating set of G. Cayley graphs associated with different generating sets are not isometric but quasi-isometric. The fundamental question which we will try to address in this book is: If G,G ′ are quasi-isometric groups, to which extent G and G ′ share the same algebraic properies?

. We determine the complexity of torsion-freeness of finitely presented groups in Kleene's arithmetical hierarchy as \Pi 0 2 -complete. This implies in particular that there is no effective listing of all torsion-free finitely presented groups, or of all non-torsion-free finitely presented groups. 0. Introduction. One way of describing a group G is to give its presentation, i.e., to write G as G = hx i (i 2 I) j Ri (where fx i j i 2 Ig is a set of "generators" and R (the set of "relators") is a set of words in fx i ; x \Gamma1 i j i 2 Ig such that G ¸ = F=H where F is the free group generated by fx i j i 2 Ig and H is the normal subgroup of F generated by R. If we can find a free group F of finite rank and a finite set of relators R, then we call G a finitely presented group. Groups arising in applications, such as fundamental groups in topology, often are given naturally via their presentations. Unfortunately, a finite presentation does not yield very good information about ...

At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an exposition of modular

In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set

Abstract. To determine if two given lists of numbers are the same set, we would sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms for equivalences arise in graph isomorphism and its variants, and the equality of permutation groups given by generators. To determine if two given graphs are cospectral (have the same eigenvalues), however, we compute their characteristic polynomials and see if they are the same; the characteristic polynomial is a complete invariant for the equivalence relation of cospectrality. This is weaker than a canonical form, and it is not known whether a polynomial-time canonical form for cospectrality exists. Note that it is a priori possible for an equivalence relation to be decidable in polynomial time without either a complete invariant or canonical form. Blass and Gurevich (“Equivalence relations, invariants, and normal forms, I and II”, 1984) ask whether these conditions on equivalence relations – having an FP canonical form, having an FP complete invariant, and simply being in P – are in fact different. They showed that this question requires non-relativizing techniques to resolve. Here we extend their results, and give new connections to probabilistic and quantum computation.