Leírás
Given a finitely presented group, many algorithmic questions arise, most importantly the solvability and complexity of the word problem. In the past few decades, geometric approaches have been developed and have proven effective in studying these, resulting in the rich theory of geometric group theory. A key factor in the development of the area was the notion of hyperbolic groups, introduced by Gromov in 1987. The notion represented a revolution in group theory due to a conjugation of factors: they can be characterized using a geometric property of their Cayley graphs, called Rips condition; they have excellent algorithmic properties: they are biautomatic (in particular, they have an effectively solvable word problem), their geodesics constitute an automatic structure.
In this series of two talks, we first introduce the basics of geometric group theory, mostly focusing on hyperbolic groups and their properties. In the second talk, we proceed to describe how the notion of hyperbolicity can be generalzied to inverse semigroups in a way that both the geometric interpretation and (some) algorithmic properties are preserved. We will briefly describe possible applications to the open problem of one-relator inverse monoids.