Gromov norm

Target entity: Gromov norm
Context triple: [Thurston norm, relatedInvariant, Gromov norm]

• A. Thurston norm
  The Thurston norm is a topological invariant on the second homology of a 3-manifold that measures the minimal complexity (via Euler characteristic) of embedded surfaces representing a given homology class.
• B. Gromov’s systolic inequality
  Gromov’s systolic inequality is a fundamental result in Riemannian geometry that bounds the volume of a manifold from below in terms of the length of its shortest non-contractible loop (the systole).
• C. Gromov compactness theorem
  The Gromov compactness theorem is a fundamental result in symplectic geometry and geometric analysis that provides compactness for families of pseudoholomorphic curves (or Riemannian manifolds with bounded geometry) up to bubbling and degeneration.
• D. Cheeger–Gromov compactness theorem
  The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
• E. Dehn function
  The Dehn function is a mathematical tool in geometric group theory that measures the complexity of filling loops with discs in a space or group, quantifying the difficulty of solving the word problem.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Gromov norm
Target entity description: The Gromov norm is a topological invariant of manifolds that measures the “size” of their fundamental class via the minimal ℓ¹-norm of singular chains representing it, closely linked to simplicial volume and geometric structures.

• A. Thurston norm
  The Thurston norm is a topological invariant on the second homology of a 3-manifold that measures the minimal complexity (via Euler characteristic) of embedded surfaces representing a given homology class.
• B. Gromov’s systolic inequality
  Gromov’s systolic inequality is a fundamental result in Riemannian geometry that bounds the volume of a manifold from below in terms of the length of its shortest non-contractible loop (the systole).
• C. Gromov compactness theorem
  The Gromov compactness theorem is a fundamental result in symplectic geometry and geometric analysis that provides compactness for families of pseudoholomorphic curves (or Riemannian manifolds with bounded geometry) up to bubbling and degeneration.
• D. Cheeger–Gromov compactness theorem
  The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
• E. Dehn function
  The Dehn function is a mathematical tool in geometric group theory that measures the complexity of filling loops with discs in a space or group, quantifying the difficulty of solving the word problem.
• F. None of above. chosen