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Let <a1,..., a m | wn> be a presentation of a group G, where w is freely and cyclically reduced and n ≥ 2 is maximal. We define a system of codimension-1 subspaces in the universal cover, and invoke a construction essentially due to Sageev to define an action of G on a CAT(0) cube complex. By proving easily formulated geometric properties of the codimension-1 subspaces we show that when n ≥ 4 the action is proper and cocompact, and that the cube complex is finite dimensional and locally finite. We also prove partial results when n = 2 or n = 3. It is also shown that the subgroups of G generated by non-empty proper subsets of {a1, a 2,..., am} embed by isometries into the whole group.

We produce a nonpositively curved square complex, X, containing exactly four squares. Its universal cover, X̃ ≅ T4 x T 4, is isomorphic to the product of two 4-valent trees. The group, pi1X, is a lattice in Aut (X̃) but π1X is not virtually a nontrivial product of free groups. There is no such example with fewer than four squares. The main ingredient in our analysis is that X̃ contains an "anti-torus" which is a certain aperiodically tiled plane.

A 2-dipath k-colouring of an oriented graph G is an assignment of k colours, 1,2, . . . , k, to the vertices of G such that vertices joined by a directed path of length two are assigned different colours. The 2-dipath chromatic number is the minimum number of colours needed in such a colouring. There are two possible models, depending on whether adjacent vertices must also be assigned different colours. For both models of 2-dipath colouring we develop the basic theory, including characterizing the oriented graphs that can be 2-dipath coloured using a small number of colours, finding bounds on the 2-dipath chromatic number, determining the complexity of deciding the existence of a 2-dipath k-colouring, describing a homomorphism model, and showing how to determine the 2-dipath chromatic number of tournaments and bipartite tournaments. / Graduate

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.

We introduce a weighted model of random trees and analyze the asymptotic properties of their heights. Our framework encompasses most trees of logarithmic height that were introduced in the context of the analysis of algorithms or combinatorics. This allows us to state a sort of "master theorem" for the height of random trees, that covers binary search trees (Devroye, 1986), random recursive trees (Devroye, 1987; Pittel, 1994), digital search trees (Pittel, 1985), scale-free trees (Pittel, 1994; Barabasi and Albert, 1999), and all polynomial families of increasing trees (Bergeron et al., 1992; Broutin et al., 2006) among others. Other applications include the shape of skinny cells in geometric structures like k-d and relaxed k-d trees. / This new approach sheds new light on the tight relationship between data structures like trees and tries that used to be studied separately. In particular, we show that digital search trees and the tries built from sequences generated by the same memoryless source share the same stable core. This link between digital search trees and tries is at the heart of our analysis of heights of tries. It permits us to derive the height of several species of tries such as the trees introduced by de la Briandais (1959) and the ternary search trees of Bentley and Sedgewick (1997). / The proofs are based on the theory of large deviations. The first order terms of the asymptotic expansions of the heights are geometrically characterized using the Crame'r functions appearing in estimates of the tail probabilities for sums of independent random variables.

Thesis (MSc)--University of Stellenbosch, 2007. / Bibliography. Also available via the Internet.

Licentiatavhandling Uppsala, Univ : 2004.

Thesis (Ph. D.) - University of Washington. / Bibliography: ℓ.[184]-185.