# Geometric Kernels of Proper Maps Between Non-Compact Surfaces

## TL;DR

This paper investigates conditions under which degree-one maps between non-compact surfaces have a geometric kernel, using Brown's proper fundamental group to extend known results from compact to non-compact surfaces.

## Contribution

It provides a sufficient condition for the existence of geometric kernels in non-compact surfaces using proper fundamental groups and characterizes conjugacy classes for this purpose.

## Key findings

- Established a sufficient condition for geometric kernels in non-compact surfaces.
- Characterized conjugacy classes in the proper fundamental group relevant to geometric kernels.
- Extended results from compact to non-compact surface mappings.

## Abstract

A map between connected $2$-manifolds has a geometric kernel if it sends a non-contractible simple loop to a null-homotopic loop. While every non-$\pi_1$-injective map between compact surfaces admits a geometric kernel, this generally fails for compact bordered or non-compact surfaces. In this paper, we use Brown's proper fundamental group to give a sufficient condition under which a degree-one map between non-compact surfaces admits a geometric kernel. Furthermore, we characterize conjugacy classes in the proper fundamental group and use this characterization to establish sufficient conditions for the existence of geometric kernels.