# Incompressible solvable representations of surface groups

## TL;DR

This paper demonstrates that for most surface groups, there exist representations into torsion-free upper-triangular matrix groups where no simple closed curve is mapped to the identity, revealing new insights into surface group representations.

## Contribution

It constructs specific representations of surface groups into torsion-free upper-triangular groups with no simple loop in the kernel, a novel result in geometric group theory.

## Key findings

- Existence of such representations for all surfaces except the projective plane and Klein bottle.
- No simple closed curve is mapped to the identity in these representations.
- Provides new examples of surface group representations with particular kernel properties.

## Abstract

The fundamental group of every surface that is not the projective plane or Klein bottle has a representation to a torsion-free group of upper-triangular matrices in SL(2,R) with no simple loop (i.e. a nontrivial element representing a simple closed curve) in the kernel.