Skip to main content
MathOverflow

Return to Question

added 392 characters in body
Source Link
Fredy
Fredy
  • 502
  • 3
  • 10

A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.

This statement is from 3-Manifold Groups, page 183-Manifold Groups, page 18 (the link is editted) by Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, it seems that the three references in the book toward this statement only give partial results (when the boundary components are already cusps).

Edits: The precise statement in my opinion should be:

Let $M$ be a compact three dimensional manifold with incompressible toroidal boundary (possibly none). If the interior of $M$ admits a hyperbolic structure, then $M$ either has finite volume, or is homeomorphic to $T^2\times I$. 

Thanks for any solutions or hints.

A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.

This statement is from 3-Manifold Groups, page 18 by Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, it seems that the three references in the book toward this statement only give partial results (when the boundary components are already cusps). Thanks for any solutions or hints.

A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.

This statement is from 3-Manifold Groups, page 18 (the link is editted) by Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, it seems that the three references in the book toward this statement only give partial results (when the boundary components are already cusps).

Edits: The precise statement in my opinion should be:

Let $M$ be a compact three dimensional manifold with incompressible toroidal boundary (possibly none). If the interior of $M$ admits a hyperbolic structure, then $M$ either has finite volume, or is homeomorphic to $T^2\times I$. 

Thanks for any solutions or hints.

Source Link
Fredy
Fredy
  • 502
  • 3
  • 10

Volume of hyperbolic 3-manifolds with toroidal boundary

A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.

This statement is from 3-Manifold Groups, page 18 by Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, it seems that the three references in the book toward this statement only give partial results (when the boundary components are already cusps). Thanks for any solutions or hints.