Research Statement : Daniel Groves

In the last few years, my research has been in geometric group theory, which attempts to understand groups via the spaces upon which they act. The last two decades have seen tremendous progress in our understanding of infinite discrete groups, and many longstanding open problems have been solved using geometric methods. I am particularly interested in coarse negative and non-positive curvature, in decision problems and in the logic of infinite discrete groups. I am also interested in the relationships of discrete groups to topology (low and high dimensional); to formal languages; and to analysis and K-theory, to mention a few. I will discuss the bearing of my work on these ideas and fields throughout this statement. First, I outline mymost important results in geometric group theory (I discuss my earlier work in Section 8). These results (and others) will be discussed in more detail later. In the series of papers [2, 3, 4], Martin Bridson and I proved (see Section 2):