Chapters
1. Introduction
2. Examples
3. Proof of Theorems and
4. Initial assumptions and reductions
5. Culler-Shalen theory
6. Bending characters of triangle group amalgams
7. The proof of Theorem when $F$ is a semi-fibre
8. The proof of Theorem when $F$ is a fibre
9. Further assumptions, reductions, and background material
10. The proof of Theorem when $F$ is non-separating but not a fibre
11. Algebraic and embedded $n$-gons in $X^\epsilon $
12. The proof of Theorem when $F$ separates but is not a semi-fibre and $t_1^+ + t_1^- >
0$
13. Background for the proof of Theorem when $F$ separates and $t_1^+ = t_1^-=0$
14. Recognizing the figure eight knot exterior
15. Completion of the proof of Theorem when $\Delta (\alpha , \beta ) \geq 7$
16. Completion of the proof of Theorem when $X^-$ is not a twisted $I$-bundle
17. Completion of the proof of Theorem when ${\Delta }(\alpha ,\beta )=6$ and $d = 1$
18. The case that $F$ separates but not a semi-fibre, $t_1^+ = t_1^- = 0$, $d \ne 1$, and $M(\alpha )$ is very small
19. The case that $F$ separates but is not a semi-fibre, $t_1^+ = t_1^- = 0$, $d>1$, and $M(\alpha )$ is not very small
20. Proof of Theorem
