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The cyclizer of an infinite permutation group G is the group generated by the cycles involved in elements of G, along with G itself. There is an ascending subgroup series beginning with G, where each term in the series is the cyclizer of the previous term. We call this series the cyclizer series for G. If this series terminates then we say the cyclizer length of G is the length of the respective cyclizer series. We study several innite permutation groups, and either determine their cyclizer series, or determine that the cyclizer series terminates and give the cyclizer length. In each of the innite permutation groups studied, the cyclizer length is at most 3. We also study the structure of a group that arises as the cyclizer of the innite cyclic group acting regularly on itself. Our study discovers an interesting innite simple group, and a family of associated innite characteristically simple groups.

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The central idea of coarse geometry is to focus on the properties of metric spaces which survive under deformations that change distances in a controlled way. These large scale properties, although too coarse to determine what happens locally, are nevertheless often able to capture the most important information about the structure of a space or a group. The relevant notions from coarse geometry and group theory are described in the beginning of this thesis. An overview of the cohomological characterisation of property A of Brodzki, Niblo and Wright is given, together with a proof that the cohomology theories used to detect property A are coarse invariants. The cohomological characterisation is used alongside a symmetrisation result for functions defining property A to give a new direct, more geometric proof that expanders do not have property A, making the connection between the two properties explicit. This is based on the observation that both the expander condition and property A can be expressed in terms of a coboundary operator which measures the size of the (co)boundary of a set of vertices. The rest of the thesis is devoted to the study of box spaces, including a description of the connections between analytic properties of groups and coarse geometric properties of box spaces. The construction of Arzhantseva, Guentner and Spakula of a box space of a finitely generated free group which coarsely embeds into Hilbert is the first example of a bounded geometry metric space which coarsely embeds into Hilbert space but does not have property A. This example is generalised here to box spaces of a large class of groups via a stability result for box spaces.

If G is a group acting on a set Ω, and α, β ∈ Ω, the directed graph whose vertex set is Ω and whose edge set is the orbit (α, β)^G is called an orbital graph of G. These graphs have many uses in permutation group theory. A graph Γ is said to be primitive if its automorphism group acts primitively on its vertex set, and is said to have connectivity one if there is a vertex α such that the graph Γ\{α} is not connected. A half-line in Γ is a one-way infinite path in Γ. The ends of a locally finite graph Γ are equivalence classes on the set of half-lines: two half-lines lie in the same end if there exist infinitely many disjoint paths between them. A complete characterisation of the primitive undirected graphs with connectivity one is already known. We give a complete characterisation in the directed case. This enables us to show that if G is a primitive permutation group with a locally finite orbital graph with more than one end, then G has a connectivity-one orbital graph Γ, and that this graph is essentially unique. Through the application of this result we are able to determine both the structure of G, and its action on the end space of Γ. If α ∈ Ω, the orbits of the stabiliser G_α are called the α-suborbits of G. The size of an α-suborbit is called a subdegree. If all subdegrees of an infinite primitive group G are finite, Adeleke and Neumann claim one may enumerate them in a non-decreasing sequence (m_r). They conjecture that the growth of the sequence (m_r) is extremal when G acts distance transitively on a locally finite graph; that is, for all natural numbers m the stabiliser in G of any vertex α permutes the vertices lying at distance m from α transitively. They also conjecture that for any primitive group G possessing a finite self-paired suborbit of size m there might exist a number c which perhaps depends upon G, perhaps only on m, such that m_r ≤ c(m-2)^r-1. We show their questions are poorly posed, as there exist primitive groups possessing at least two distinct subdegrees, each occurring infinitely often. The subdegrees of such groups cannot be enumerated as claimed. We give a revised definition of subdegree enumeration and growth, and show that under these new definitions their conjecture is true for groups exhibiting exponential subdegree growth above a prescribed bound.