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The symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces.

Let Mk be the category of algebras over a unique factorization domain k, and let ind-Affk denote the category of pro-representable functors from Mk to the category E of sets. It is shown that ind-Affk is a closed model category in such a way that its associated homotopy category Ho(ind-Affk) is equivalent to the homotopy category Ho(S) which comes from the category S of simplicial sets. The equivalence is induced by functors Sk: ind-Affk -> S and Rk: S-> ind-Affk. In an effort to determine what is measured by the homotopy groups πi(X) := πi. (Sk X) of X in ind-Affk in the case where k is an algebraically closed field, some homotopy groups of affine reduced algebraic groups G over k are computed. It is shown that, if G is connected, then π₀ (G) = * if and only if the group G(k) of k-rational points of G is generated by unipotents. A fibration theory is developed for homomorphisms of algebraic groups which are surjective on rational points which allows the computation of the homotopy groups of any connected algebraic group G in terms of the homotopy groups of the universal covering groups of the simple algebraic subgroups of the associated semi-simple group G/R(G), where R(G) is the solvable radical of G. The homotopy groups of simple Chevalley groups over almost all fields k are studied. It is shown that the homotopy groups of the special linear groups S1n and of the symplectic groups Sp2m converge, respectively, to the K-theory and ₋₁L-theory of the underlying field k. It is shown that there are isomorphisms π₁ (S1n ) = H₂(S1n (k);Z) = K₂(k) for n ≥ 3 and almost all fields k, and π₁ (Sp₂m ) = H₂(Sp₂m) (k);Z) = ₋₁L₂(k) for m ≥ 1 and almost all fields k of characteristic ≠ 2, where Z denotes the ring of integers. It is also shown that π₁(Sp₂m) = H₂(Sp2m(k);Z) = K₂ (k) if k is algebraically closed of arbitrary characteristic. A spectral sequence for the homology of the classifying space of a simplicial group is used for all of these calculations. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis constructs several groups entirely by hand via their symmetric presentations. In particular, the technique of double coset enumeration is used to manually construct J₃ : 2, the automorphism group of the Janko group J₃, and represent every element of the group as a permutation of PSL₂ (16) : 4, on 120 letters, followed by a word of length at most 3.

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Discusses certain one-parameter subgroups of the low-rank unitary groups called root subgroups. Unitary groups also have representations of Lie type which means they consist of transformations that act as automorphisms of an underlying Lie algebra, in this case the special linear algebra. Exploring this definition of the unitary groups, we find a correlation, via exponentiation, to the basis elements of Lie algebra.

Advantages of the double coset enumeration technique include its use to represent group elements in a convenient shorter form than their usual permutation representations and to find nice permutation representations for groups. In this thesis we construct, by hand, several groups, including U₃(3) : 2, L₂(13), PGL₂(11), and PGL₂(7), represent their elements in the short form (symmetric representation) and produce their permutation representations.

This thesis investigates the structure and properties of hyperbolic 3-manifold groups (particularly knot and link groups) and arithmetic Kleinian groups. In Chapter 2, we establish a stronger version of a conjecture of A. Reid and others in the arithmetic case: if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic (and hence peripheral). In Chapter 3, we identify all Kleinian groups that can be generated by two elements for which equality holds in Jørgensen’s Inequality in two cases: torsion-free Kleinian groups and non-cocompact arithmetic Kleinian groups. / text

Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of S∞.

Let F be a finite field of q elements and G be a primitive group on a finite set . Then there is a G-action on , namely a map G ! , (g; !) 7! !g = g!; satisfying !gg0 = (gg0)! = g(g0!) for all g; g0 2 G and all ! 2 , and that !1 = 1! = ! for all ! 2 : Let F = ff j f : ! Fg, be the vector space over F with basis . Extending the G-action on linearly, F becomes an FG-module called an FG- permutation module. We are interested in finding all G-invariant FG-submodules, i.e., codes in F . The elements f 2 F are written in the form f = P !2 a! ! where ! is a characteristic function. The natural action of an element g 2 G is given by g P !2 a! ! = P !2 a! g(!): This action of G preserves the natural bilinear form defined by * X a! !; X b! ! + = X a!b!: In this thesis a program is proposed on how to determine codes with given primitive permutation group. The approach is modular representation theoretic and based on a study of maximal submodules of permutation modules F defined by the action of a finite group G on G-sets = G=Gx. This approach provides the advantage of an explicit basis for the code. There appear slightly different concepts of (linear) codes in the literature. Following Knapp and Schmid [83] a code over some finite field F will be a triple (V; ; F), where V = F is a free FG-module of finite rank with basis and a submodule C. By convention we call C a code having ambient space V and ambient basis . F is the alphabet of the code C, the degree n of V its length, and C is an [n; k]-code if C is a free module of dimension k. In this thesis we have surveyed some known methods of constructing codes from primitive permutation representations of finite groups. Generally, our program is more inclusive than these methods as the codes obtained using our approach include the codes obtained using these other methods. The designs obtained by other authors (see for example [40]) are found using our method, and these are in general defined by the support of the codewords of given weight in the codes. Moreover, this method allows for a geometric interpretation of many classes of codewords, and helps establish links with other combinatorial structures, such as designs and graphs. To illustrate the program we determine all 2-modular codes that admit the two known non-isomorphic simple linear groups of order 20160, namely L3(4) and L4(2) = A8. In the process we enumerate and classify all codes preserved by such groups, and provide the lattice of submodules for the corresponding permutation modules. It turns out that there are no self-orthogonal or self-dual codes invariant under these groups, and also that the automorphism groups of their respective codes are in most cases not the prescribed groups. We make use of the Assmus Matson Theorem and the Mac Williams identities in the study of the dual codes. We observe that in all cases the sets of several classes of non-trivial codewords are stabilized by maximal subgroups of the automorphism groups of the codes. The study of the codes invariant under the simple linear group L4(2) leads as a by-product to a unique flag-transitive, point primitive symmetric 2-(64; 28; 12) design preserved by the affi ne group of type 26:S6(2). This has consequently prompted the study of binary codes from the row span of the adjacency matrices of a class of 46 non-isomorphic symmetric 2-(64; 28; 12) designs invariant under the Frobenius group of order 21. Codes obtained from the orbit matrices of these designs have also been studied. The thesis concludes with a discussion of codes that are left invariant by the simple symplectic group S6(2) in all its 2-modular primitive permutation representations. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.