The central aim of this thesis is to prove various Diophantine approximation results which quantify cases of the weak approximation theorem. We take finitely many different completions of a global field , take their direct product, and approximate elements of this space by elements of the global field. We prove analogues of Gallagher's zero-one law, and of the Duffin-Schaeffer theorem, in two setups of this type: the direct product of finitely many completions of Q (always including R), and the direct product of all of the Archimedean completions of a general number field . The second result in particular forms a significant improvement on existing results, which were only proven for imaginary quadratic fields.

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In this thesis we treat a number of topics related to generation of finite groups with motivation from their action on surfaces. The majority of our findings are presented in two chapters which can be read independently. The first deals with Beauville groups which are automorphism groups of the product of two Riemann surfaces with genus g > 1, subject to some further conditions. When these two surfaces are isomorphic and transposed by elements of G we say these groups are mixed, otherwise they are unmixed. We first examine the relationship between when an almost simple group and its socle are unmixed Beauville groups and then go on to determine explicit examples of several infinite families of mixed Beauville groups. In the second we determine the Mobius function of the small Ree groups 2G2(32m+1) = R(32m+1), where m >0, and use this to enumerate various ordered generating n-tuples of these groups. We then apply this to questions of the generation and asymptotic generation of the small Ree groups as well as interpretations in other categories, such as the number of regular coverings of a surface with a given fundamental group and whose covering group is isomorphic to R(32m+1).

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A (2,3,7)-group is a group generated by two elements, one an involution and the other of order 3, whose product is of order 7. By the (2,3,7)-group is meant the group which is completely defined by these relations. In this thesis we discuss the (2,3,7)-group and its homomorphisms into certain finite simple groups.

This thesis is divided in two equal parts. We start the first part by studying the Kato-spectrum of a commutative monoid M, denoted by KSpec(M). We show that the functor M → KSpec(M) is representable and discuss a few consequences of this fact. In particular, when M is additionally finitely generated, we give an efficient way of calculating it explicitly. We then move on to study the cohomology theory of monoid schemes in general and apply it to vector- and particularly, line bundles. The isomorphism class of the latter is the Picard group. We show that under some assumptions on our monoid scheme X, if k is an integral domain (resp. PID), then the induced map Pic(X) → Pic(Xk) from X to its realisation is a monomorphism (resp. isomorphism). We then focus on the Pic functor and show that it respects finite products. Finally, we generalise several important constructions and notions such as cancellative monoids, smoothness and Cartier divisors, and prove important results for them. The main results of the second part can be summed up in fewer words. We prove that for good topological spaces X the assignment U → II₁(U) is the terminal object of the 2-category of costacks. Here U is an open subset of X and II₁(U) denotes the fundamental groupoid of U. This result translates to the étale fundamental groupoid as well, though the proof there is completely different and involves studying and generalising Galois categories.

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