The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. Although a developed theory of cosets in inverse semigroups exists, it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finte index are much weaker than in the group case. Nevertheless, many aspects of this theory remain of interest, and some of them are addressed in this thesis. We study the basic theory of cosets in inverse semigroups, including an index formula for chains of subgroups and an analogue of M. Hall’s Theorem on counting subgroups of finite index in finitely generated groups. We then look at specific examples, classifying the finite index inverse subsemigroups in polycyclic monoids and in graph inverse semigroups. Finally, we look at the connection between the properties of finite generation and having finte index: these were shown to be equivalent for free inverse monoids by Margolis and Meakin.

I investigate the K_2 groups of the quotients of Fermat curves given in projective coordinates by the equation F_n:X^n+Y^n=Z^n. On any quotient where the number of known elements is equal to the rank predicted by Beilinson’s Conjecture I verify numerically that the determinant of the matrix of regulator values agrees with the leading coefficient of the L-function up to a simple rational number. The main source of K_2 elements are the so-called “symbols with divisorial support at infinity” that were found by Ross in the 1990’s. These consist of symbols of the form f, g where f and g have divisors whose points P all satisfy XY Z(P) = 0. The image of this subgroup under the regulator is computed and is found to be of rank predicted by Beilinson’s Conjecture on eleven nonisomorphic quotients of dimension greater than one. The L-functions of these quotients are computed using Dokchitser’s ComputeL package and Beilinson’s Conjecture is verified numerically to a precision of 200 decimal digits. In chapter five, with careful analysis of a certain 2 × 2 determinant it is shown that a particular hyperelliptic quotient of all the Fermat curves has K_2 group of rank at least two. In the last chapter of the dissertation, a computational method is used in order to discover new elements of K_2. These elements are rigorously proven to be tame and allow for the full verification of Beilinson’s Conjecture on the Fermat curves F_7 and F_9. Also the method allows us to verify Beilinson’s Conjecture on certain hyperelliptic quotients of F_8 and F_10. Quotients where a similar method might be successful are also suggested.

Motivated by homological mirror symmetry, Smith and Thomas tried to construct a link invariant considering the derived category of coherent sheaves on the Hilbert scheme of n points on the minimal resolution of the Klenian singularity of type A, and an object L(n) thereof. The braid group acts on this category by spherical twists, so one obtains a braid invariant by taking the Ext between L(n) and its image under the braid group action. Smith and Thomas proved that taking the plat closure of the braid, this cohomology does not produce a link invariant but is close to doing so, and they conjectured that, in order to fix the one knot relation that is not satisfied, one has to consider a deformation of the Hilbert scheme. In this thesis, we give a non-commutative approach to this problem: the commutative picture can be quantised by considering modules for the rational Cherednik algebra of cyclotomic type. This algebra gives a quantisation of the Hilbert scheme and there is a localisation theorem which allows one to work in the algebraic setting. In this context, the role of L(n) turns out to be played by a certain module for the rational Cherednik algebra which we define for k=0. We then show that this module deforms to non-zero values of k. There is an action of the braid group on the derived category of category ? by twisting functors, which is defined at all deformation parameters, whereas the existence of the action on deformed Hilbert schemes in the commutative setting has not been rigorously established. We prove the analogue of the Smith-Thomas theorem, and conjecture that the braid invariant given by the algebraic analogue of the Smith-Thomas construction yields a link invariant for certain non-zero values of the deformation parameter.

This thesis concerns the diameter and spectral gap of finite groups. Our focus shall be on the asymptotic behaviour of these quantities for sequences of finite groups arising as quotients of a fixed infinite group. In Chapter 3 we give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-p domain (with exponent depending on the dimension); Chevalley groups over a pro-p domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups. In Chapter 4 we construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of SL2(Fp[t]) modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in SL2(Fp[t]) Finally, in Chapter 5 we produce new expander congruence quotients of SL2 (Zp), generalising work of Bourgain and Gamburd. The proof combines the Solovay-Kitaev procedure with a quantitative analysis of the algebraic geometry of these groups, which in turn relies on previously known examples of expanders.

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In this thesis we investigate the notion of a subgroup of a quantum group. We suggest a general definition, which takes into account the work that has been done for quantum homogeneous spaces. We further restrict our attention to reductive subgroups, where some faithful flatness conditions apply. We give examples of quantum subgroups, some known and some new, which are all part of the family of spherical subgroups. The ultimate goal would be to quantize all spherical subgroups. Furthermore, we proceed with a categorical approach to the problem of finding quantum subgroups. We translate all existing results into the language of module and monoidal categories and give another characterization of the notion of a quantum subgroup.

Groups naturally occu as the symmetries of an object. This is why they appear in so many different areas of mathematics. For example we find class grops in number theory, fundamental groups in topology, and amenable groups in analysis. In this thesis we will use techniques and approaches from various fields in order to study groups. This is a 'three paper' thesis, meaning that the main body of the document is made up of three papers. The first two of these look at permutation groups which contain all permutations with finite support, the first focussing on decision problems and the second on the R? property (which involves counting the number of twisting conjugacy classes in a group). The third works with wreath products C}Z where C is cyclic, and looks to dermine the probability of choosing two elements in a group which commute (known as the degree of commutativity, a topic which has been studied for finite groups intensely but at the time of writing this thesis has only two papers involving infinite groups, one of which is in this thesis).