The fundamental group of a two-dimensional complex with the fixed point property

JONATHAN ARIEL BARMAK; IVÁN SADOFSCHI COSTA

Barranquilla

Congreso; V Congreso latinoamericano de matemáticos; 2016

By the Lefschetz fixed point theorem, if a finite group G admits a presentation with the same number of relators as generators, then there exists a compact polyhedron X with fundamental group isomorphic to G and the fixed point property (i.e. each self map of X has a fixed point). For groups with non-zero deficiency the problem is related to a question posed in 1969 by R.H. Bing. In this talk I will show that non-cyclic abelian groups do not appear as fundamental groups of two-dimensional complexes with the fixed point property. An example by I. Sadofschi Costa of a group presented by two generators and three relators provides the first example of a 2-complex with the fixed point property and non-trivial reduced Euler Characteristic, answering Bing?s question in the affirmative.