**Mahan Maharaj**

Belur University, West Bengal

December 5, 2012

**Low Dimensional Projective Groups**:
We shall discuss some recent progress towards the conjecture that
if the fundamental group G of a compact projective manifold has
cohomological dimension less than 4, it must be the fundamental group of a Riemann surface.
Sample theorems include:
a) Let $1 -> N -> G -> Q -> 1$ be an exact sequence of finitely presented groups, where $Q$ is infinite and not virtually
cyclic, and is the fundamental group of some closed 3-manifold. If $G$ is Kahler, we show that $Q$ contains as a finite index subgroup
either a finite index subgroup of the 3-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. It follows that no infinite 3-manifold group can be Kaehler(originally proved by Dimca and Suciu).
b) If $G$ is a one-relator group, it must be the fundamental group of a Riemann surface.
c) If $G$ has cohomological dimension 2, then modulo the Shafarevich
conjecture, it must be the fundamental group of a Rie