Gromov norm

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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.

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. Thurston also used the Gromov norm to prove that hyperbolic volume decreases under hyperbolic Dehn surgery.