Abstract ping-pong systems in subgroups of Thompson’s groups

In this thesis we study abstract ping-pong systems of homeomorphisms which admit either linear or cyclic orders. Abstract ping-pong systems on sets of permutations were defined by Bleak, Brin, Kassabov, Moore, and Zaremsky as an adaptation of the classical ping-pong lemma; in particular, sets which admit abstract ping-pong systems need not generate free groups. We first focus on linearly ordered finite abstract ping-pong systems within PL₊(I), the group of piecewise-linear orientation-preserving homeomorphisms of the closed unit interval. It is known that such sets generate groups which embed into Thompson’s group F. We discuss PL₊(I)-transition chains in general, and prove a collection of results sufficient to deduce that a group which admits chains with certain properties must also admit a subgroup isomorphic to Thompson’s group F. We then analyse the dynamical properties of a family of elementary amenable groups defined by Bleak, Brin, and Moore, and prove a foundational case of a conjecture of Bleak. Secondly, we consider an infinite family of finite abstract ping-pong systems for homeomorphisms of S¹ equipped with cyclic - but not linear - orders. These sets generate the n-ring groups, defined by Hyde, Lodha, and Rivas. The n-ring groups are known to embed into Thompson’s group T, and to have finitely generated, simple, left orderable commutator subgroups. We are interested in the question of the finite presentability of these groups. We provide an algorithm which generates a recursive presentation for each of these groups, but do not settle this question. We investigate two suggested paths to a proof and demonstrate where they fail.