  We associate to each finite presentation of a group G a compact CW-complex
that is a 3-manifold in the complement of a point, and whose fundamental group
is isomorphic to G. We use this complex to define a notion of genus for G and
give examples, and also define a notion of `closed group'. A group has genus 0
if and only if it is the fundamental group of a compact orientable 3-manifold.
