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Reversible and other generalised torsion elements in Seifert-fibered spaces

Anushree Das, Debattam Das

TL;DR

The paper addresses the problem of classifying reversible elements in fundamental groups arising from surfaces and Seifert-fibered spaces, and extends the analysis to generalized torsion phenomena. It develops a framework based on Fuchsian group structure and axis geometry to characterize reversibility, with the classification dependent on the classifying homomorphism in Seifert-fibered spaces. Key contributions include explicit reversible-element forms for orientable and non-orientable orbit surfaces, application of these results to the universal extension of Fuchsian groups, and construction of generalized $3$-torsion elements in PSL(2, Z) and in the braid group $B_3$. The work connects algebraic reversibility to geometric features like even-order vertices and Klein bottle embeddings, providing tools for understanding central extensions of Fuchsian groups and their braid-group realizations.

Abstract

An element $a$ in a group $Γ$ is called \emph{reversible} if there exists $g \in Γ$ such that $gag^{-1}=a^{-1}$. The reversible elements are also known as `real elements' or `reciprocal elements' in literature. In this paper, we classify the reversible elements in Fuchsian groups, and use this classification to find all reversible elements in a Seifert-fibered group. We then apply the classification to the braid groups, particularly to the braid group on $3$ strands. We further study generalised 3-torsion elements in PSL(2,Z), and use this to analyse the existence of generalised 3-torsion elements in Seifert-fibered spaces in general, and braid groups on 3 strands in particular.

Reversible and other generalised torsion elements in Seifert-fibered spaces

TL;DR

The paper addresses the problem of classifying reversible elements in fundamental groups arising from surfaces and Seifert-fibered spaces, and extends the analysis to generalized torsion phenomena. It develops a framework based on Fuchsian group structure and axis geometry to characterize reversibility, with the classification dependent on the classifying homomorphism in Seifert-fibered spaces. Key contributions include explicit reversible-element forms for orientable and non-orientable orbit surfaces, application of these results to the universal extension of Fuchsian groups, and construction of generalized -torsion elements in PSL(2, Z) and in the braid group . The work connects algebraic reversibility to geometric features like even-order vertices and Klein bottle embeddings, providing tools for understanding central extensions of Fuchsian groups and their braid-group realizations.

Abstract

An element in a group is called \emph{reversible} if there exists such that . The reversible elements are also known as `real elements' or `reciprocal elements' in literature. In this paper, we classify the reversible elements in Fuchsian groups, and use this classification to find all reversible elements in a Seifert-fibered group. We then apply the classification to the braid groups, particularly to the braid group on strands. We further study generalised 3-torsion elements in PSL(2,Z), and use this to analyse the existence of generalised 3-torsion elements in Seifert-fibered spaces in general, and braid groups on 3 strands in particular.
Paper Structure (9 sections, 21 theorems, 46 equations)

This paper contains 9 sections, 21 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Reversiblity in Fundamental groups of the surfaces
  3. Reversible classes of Fuchsian group
  4. Proof of Theorem \ref{['thm3']}
  5. Reversible classes of non-orientable surface fundamental groups
  6. Seifert fibered space
  7. Proof of Theorem \ref{['l1']}
  8. Application to universal extension of Fuchsian groups
  9. Generalised 3-Torsion elements in $\pi_{1}(B_{3})$

Key Result

Theorem 1.1

If a Seifert fibered space $M$ has an orientable orbit surface. If $\phi$ is nontrivial, the reversible elements of $\pi_1(M)$ are the following: If $\phi$ is trivial, $\pi_1(M)$ has reversible elements as the elements conjugate to $c_i^{\frac{\mu_i}{2}}kc_j^{\frac{-\mu_j}{2}}k^{-1}$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Corollary 3.1
  • Lemma 3.2
  • Corollary 3.3
  • ...and 16 more