RESEARCH STATEMENT : ERIC SWARTZ 1 Introduction and Previous Work

First and foremost, I have extensive experience with permutation group theory. There is an extensive literature on groups that act primitively on a set Ω, i.e., those groups which do not preserve a nontrivial system of blocks. Furthering this is the notion of quasiprimitivity. A group G acts quasiprimitively on a set Ω if every nontrivial normal subgroup of G is transitive on Ω. Every group acting primitively on a set also acts quasiprimitively, so it is a natural generalization. On the other hand, it is in certain respects a more natural notion. For instance, given a graph Γ with group of automorphisms G, for any normal subgroup N of G one can define a (normal) quotient graph ΓN , where the vertices of ΓN are N -orbits of vertices of Γ, and two vertices Σ1 and Σ2 of ΓN (which, again, correspond to N -orbits of vertices of Γ) are adjacent if and only if there are vertices α ∈ Σ1 and β ∈ Σ2 such that α and β are adjacent in Γ. The reason that this notion of quotient graph is particularly useful is that quotients inherit properties of the original graph. With this in mind, an s-arc of a graph Γ is a sequence of s+1 vertices (α0, α1, . . . , αs) such that αi is adjacent to αi+1 for 0 ≤ i ≤ s and αj−1 6= αj+1 for 1 ≤ j ≤ s − 1. (Note that this means that vertices can be repeated in the sequence, potentially, just not immediately.) A graph Γ is said to be s-arc transitive if, given any two s-arcs in Γ, there exists an automorphism of the graph taking one to the other. A graph Γ is said to be locally s-arc transitive if for any fixed vertex α and two s-arcs starting at α, there is an automorphism mapping one to the other. While s-arc transtitivity implies local s-arc transitivity, the converse is not true: as an example, the complete bipartite graph K2,3 is locally 3-arc transitive but not even vertex transitive. Concepts such as s-arc transitivity and local s-arc transitivity are preserved under the process of taking normal quotients. One can now divide graphs into two categories: basic graphs, i.e., those which have no nontrivial quotient, and regular covers of these basic graphs. (Naturally, graphs with a group G acting quasiprimitively with respect to one or two orbits of vertices are basic in this context.) For instances of my work in this area, see [19] and [15]. In further work that is in preparation, Luke Morgan, Gabriel Verret, and I have shown that, given any integers k and n, for sufficiently large d (depending only on n) there are only finitely many 2-arc transitive graphs of degree d and order kp, where p can be any prime.