Uniqueness theorem for completely non-degenerate B-groups

TL;DR

This paper proves that completely non-degenerate B-groups are uniquely identified by their factors, establishing a conformal equivalence criterion and analyzing the factor as a triple, with implications for related groups.

Contribution

It introduces a uniqueness theorem for completely non-degenerate B-groups based on their factors and studies the factor as a structured triple.

Findings

Completely non-degenerate B-groups are uniquely determined by their factors.

Conformally equivalent factors imply M"obius conjugacy for these groups.

The factor of a B-group can be characterized as a triple involving the main factor, characteristic complex, and a homotopy class.

Abstract

We prove that a completely non-degenerate B-group is uniquely determined by its factor: two such groups with conformally equivalent factors are M\"obius conjugate. A similar property is inherent to the quasi-Fuchsian groups but not to degenerate B-groups. We also study the factor of a B-group as a triple: the main factor, the marked characteristic complex, and a homotopy class of maps of the first to the second one.