Gromov norm

In mathematics, the Gromov norm (or simplicial volume) of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class.^[1][2]

It is named after Mikhail Gromov, who with William Thurston, proved that the Gromov norm of a finite volume hyperbolic n-manifold is proportional to the hyperbolic volume.^[1] Thurston also used the Gromov norm to prove that hyperbolic volume decreases under hyperbolic Dehn surgery.^[3]

References

1. Benedetti, Riccardo; Petronio, Carlo (1992), Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, p. 105, doi:10.1007/978-3-642-58158-8, ISBN 3-540-55534-X, MR 1219310.
2. Ratcliffe, John G. (2006), Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149 (2nd ed.), Berlin: Springer, p. 555, ISBN 978-0387-33197-3, MR 2249478.
3. Benedetti & Petronio (1992), pp. 196ff.

External links

• Simplicial volume at the Manifold Atlas.