Artin presentations, triangle groups, and 4-manifolds

Abstract

González-Acuña showed that Artin presentations characterize closed, orientable 3-manifold groups. Winkelnkemper later discovered that each Artin presentation determines a smooth, compact, simply connected 4-manifold. We utilize triangle groups to find all Artin presentations on two generators that present the trivial group. We then determine all smooth, closed, simply connected 4-manifolds with second betti number at most two that appear in Artin presentation theory.