Conservative Subgroup Separability For Surfaces With Boundary

TL;DR

This paper proves that for surfaces with boundary, certain subgroups can be separated from other elements via finite covers that preserve boundary components, enhancing understanding of subgroup separability in surface groups.

Contribution

It establishes a conservative subgroup separability result for surfaces with boundary, showing finite covers can distinguish specific subgroups while maintaining boundary structure.

Findings

Subgroups without peripheral elements are separable from finitely many elements.

Finite covers can be constructed to preserve boundary components.

The result applies to finitely generated subgroups of surface groups.

Abstract

If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = {\pi}_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .