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Circular orderability of 3-manifold groups

Idrissa Ba, Adam Clay

Abstract

This paper initiates the study of circular orderability of $3$-manifold groups, motivated by the L-space conjecture. We show that a compact, connected, $\mathbb{P}^2$-irreducible $3$-manifold has a circularly orderable fundamental group if and only if there exists a finite cyclic cover with left-orderable fundamental group, which naturally leads to a "circular orderability version" of the L-space conjecture. We also show that the fundamental groups of almost all graph manifolds are circularly orderable, and contrast the behaviour of circularly orderability and left-orderability with respect to the operations of Dehn surgery and taking cyclic branched covers.

Circular orderability of 3-manifold groups

Abstract

This paper initiates the study of circular orderability of -manifold groups, motivated by the L-space conjecture. We show that a compact, connected, -irreducible -manifold has a circularly orderable fundamental group if and only if there exists a finite cyclic cover with left-orderable fundamental group, which naturally leads to a "circular orderability version" of the L-space conjecture. We also show that the fundamental groups of almost all graph manifolds are circularly orderable, and contrast the behaviour of circularly orderability and left-orderability with respect to the operations of Dehn surgery and taking cyclic branched covers.

Paper Structure

This paper contains 15 sections, 37 theorems, 48 equations, 2 figures.

Table of Contents

  1. introduction
  2. Left and circular orderability
  3. Fundamental groups of $3$-manifolds and orderability
  4. Slope detection and $3$-manifolds having nontrivial JSJ decomposition
  5. Rational longitudes and knot manifolds
  6. Seifert fibred manifolds and graph manifolds
  7. Seifert fibred manifolds
  8. Graph manifolds
  9. Graph manifolds with two pieces
  10. cyclic branched covers and Dehn surgery
  11. Cyclic branched covers
  12. Double branched covers
  13. Generalized Fibonacci groups
  14. Generalized Takahashi manifolds
  15. Dehn surgery

Key Result

Theorem 1.1

Suppose that $M$ is a compact, connected, $\mathbb{P}^2$-irreducible $3$-manifold. Then $\pi_1(M)$ is circularly orderable if and only if $M$ admits a finite cyclic cover with left-orderable fundamental group.

Figures (2)

  • Figure 1: Example of a splice diagram.
  • Figure 2: The braid that defines the generalized Takahashi manifold $T_{n,m}(\frac{p_{k,j}}{q_{k,j}}; \frac{r_{k,j}}{s_{k,j}})$. The fraction used to label each box determines the rational tangle used in that box to create $L_{n,m}(\frac{p_{k,j}}{q_{k,j}}; \frac{r_{k,j}}{s_{k,j}})$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • proof
  • ...and 64 more