In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups. / Science, Faculty of / Mathematics, Department of / Graduate

The goal of this paper is to explore the first cohomology group of groups G that are not necessarily finitely generated. Our focus is on l^p-cohomology, 1 leq p leq infty, and what results regarding finitely generated groups change when G is infinitely generated. In particular, for abelian groups and locally finite groups, the l^p-cohomology is non-zero when G is countable, but vanishes when G has sufficient cardinality. We then show that the l^infty-cohomology remains unchanged for many classes of groups, before looking at several results regarding the injectivity of induced maps from embeddings of G-modules. We present several new results for countable groups, and discuss which results fail to hold in the general uncountable case. Lastly, we present results regarding reduced cohomology, including a useful lemma extending vanishing results for finitely generated groups to the infinitely generated case. / Ph. D.

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Doctor of Philosophy / Department of Mathematics / John S. Maginnis / In this work, we find the module structure of the cohomology of the group of four by four upper triangular matrices (with ones on the diagonal) with entries from the field on three elements modulo its center. Some of the relations amongst the generators for the cohomology ring are also given. This cohomology is found by considering a certain split extension. We show that the associated Lyndon-Hochschild-Serre spectral sequence collapses at the second page by illustrating a set of generators for the cohomology ring from generating elements of the second page. We also consider two other extensions using more traditional techniques. In the first we introduce some new results giving degree four and five differentials in spectral sequences associated to extensions of a general class of groups and apply these to both the extensions.

We want to take a look at the first cohomology group H^1(G, l^2(G)), in particular when G is locally-finite. First, though, we discuss some results about the space H^1(G, C G) for G locally-finite, as well as the space H^1(G, l^2(G)) when G is finitely generated. We show that, although in the case when G is finitely generated the embedding of C G into l^2(G) induces an embedding of the cohomology groups H^1(G, C G) into H^1(G, l^2(G)), when G is countably-infinite locally-finite, the induced homomorphism is not an embedding. However, even though the induced homomorphism is not an embedding, we still have that H^1(G, l^2(G)) neq 0 when G is countably-infinite locally-finite. Finally, we give some sufficient conditions for H^1(G,l^2(G)) to be zero or non-zero. / Master of Science

The cohomology ring of a finite cyclic group was explicitly computed by Cartan and Eilenberg in their 1956 book on Homological Algebra. It is surprising that the cohomology ring for the next simplest example, that of a finite abelian group, has still not been treated in a systematic way. The results that we do have are combinatorial in nature and have been obtained using "brute force" computations. In this thesis we will give a systematic method for computing the cohomology ring of a finite abelian group. A major ingredient in this treatment will be the Tate resolution of a commutative ring R (with trivial group action) over the group ring RG, for some finite abelian group G. Using the Tate resolution we will be able to compute the cohomology ring for a finite cyclic group, and confirm that this computation agrees with what is known from Cartan-Eilenberg. Then we will generalize this technique to compute the cohomology ring for a finite abelian group. The presentation we will give is simpler than what is in the literature to date. We will then see that a straightforward generalization of the Tate resolution from a group ring to an arbitrary ring defined by monic polynomials will yield a method for computing the Hochschild cohomology algebra of that ring. In particular we will re-prove some results from the literature in a much more unified way than they were originally proved. We will also be able to prove some new results.

The cohomology ring of a finite cyclic group was explicitly computed by Cartan and Eilenberg in their 1956 book on Homological Algebra. It is surprising that the cohomology ring for the next simplest example, that of a finite abelian group, has still not been treated in a systematic way. The results that we do have are combinatorial in nature and have been obtained using "brute force" computations. In this thesis we will give a systematic method for computing the cohomology ring of a finite abelian group. A major ingredient in this treatment will be the Tate resolution of a commutative ring R (with trivial group action) over the group ring RG, for some finite abelian group G. Using the Tate resolution we will be able to compute the cohomology ring for a finite cyclic group, and confirm that this computation agrees with what is known from Cartan-Eilenberg. Then we will generalize this technique to compute the cohomology ring for a finite abelian group. The presentation we will give is simpler than what is in the literature to date. We will then see that a straightforward generalization of the Tate resolution from a group ring to an arbitrary ring defined by monic polynomials will yield a method for computing the Hochschild cohomology algebra of that ring. In particular we will re-prove some results from the literature in a much more unified way than they were originally proved. We will also be able to prove some new results.

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