Higher dimensional Thompson groups

Three groups F ⊆ T ⊆ V , known as Thompson groups, have generated interest since R. J. Thompson introduced them in the late 1960s. Part of their initial interest was the fact that T and V supplied the first known examples of infinite, simple, finitely presented groups. Since then, other properties of the groups have been studied as well as their interaction with other areas of mathematics. The standard reference for the groups F ⊆ T ⊆ V is [7]. It is not necessary to have [7] in hand while reading this paper, but it would not hurt. The largest group V can be described as a subgroup of the homeomorphism group of the Cantor set C. Intrinsic to this description is the standard “deleted middle thirds” construction of the Cantor set as a subset of the unit interval. Since the unit interval is a 1-dimensional object, we will refer to V as a 1-dimensional Thompson group. In this paper, we describe an intrinsically 2-dimensional group 2V that is more naturally described as a subgroup of the homeomorphism group of C ×C. We will show in this paper that 2V is infinite, simple and finitely generated, and in another paper [4], that 2V is finitely presented. In spite of the fact that C × C is homeomorphic to C, we will also show in this paper that 2V is not isomorphic to V . In fact, we will show that 2V is not isomorphic to any group in a list of other infinite, simple, finitely presented groups that are closely related to T and V . However, we do not show that 2V is not isomorphic to all known infinite, simple, finitely presented groups.