Volume rigidity

Volume rigidity (ger .: volume rigidity ) refers to two different concepts in mathematics .

Volume rigidity according to Thurston and Besson-Courteois-Gallot

Theorem (Thurston): If there is a continuous mapping between complete hyperbolic manifolds of finite volume of the dimension , then applies to the mapping degree ${\displaystyle f\colon M_{1}\to M_{2}}$ ${\displaystyle n\geq 3}$

${\displaystyle deg(f)\leq {\frac {Vol(M_{1})}{Vol(M_{2})}}}$

and equality only when it is actually homotopic to a Riemannian superposition . ${\displaystyle f}$

Theorem (Besson-Courteois-Gallot, Boland-Connell-Souto): If a continuous mapping between complete Riemannian manifolds of finite volume of dimension with ${\displaystyle f\colon (M_{1},g_{1})\to (M_{2},g_{2})}$ ${\displaystyle n\geq 3}$

${\displaystyle Ric(g_{1})\geq -(n-1)g_{1},-a\leq K(g_{2})\leq -1}$

is for a constant , then applies to the degree of mapping ${\displaystyle a\geq 1}$

${\displaystyle deg(f)\leq {\frac {Vol(M_{1})}{Vol(M_{2})}}}$

and equality only when it is actually homotopic to a Riemannian superposition. ${\displaystyle f}$

Volume rigidity according to Goldman

Theorem (Goldman): Let be a closed hyperbolic surface and a representation . Then applies ${\displaystyle \Sigma _{g}}$ ${\displaystyle \rho \colon \pi _{1}\Sigma _{g}\to Isom^{+}(\mathbb {H} ^{2})=PSL(2,\mathbb {R} )}$

${\displaystyle vol(\rho )\leq vol(\Sigma _{g})}$

and equality only when is discreet and loyal . ${\displaystyle \rho }$

Theorem (Dunfield, Francaviglia-Klaff): Let a hyperbolic manifold of finite volume of the dimension and a representation. Then applies ${\displaystyle M}$ ${\displaystyle n\geq 3}$ ${\displaystyle \rho \colon \pi _{1}M\to Isom^{+}(\mathbb {H} ^{n})=SO(n,1)_{0}}$

${\displaystyle vol(\rho )\leq vol(M)}$

and equality only when is discreet and loyal. ${\displaystyle \rho }$

Theorem (corollary to the superstariness theorem ): Let be a compact locally symmetric space of non-compact type with no - or - factor, and be a representation. Then applies ${\displaystyle M=\Gamma \backslash G/K}$ ${\displaystyle G}$ ${\displaystyle SO(n,1)}$ ${\displaystyle SU(n,1)}$ ${\displaystyle \rho \colon \Gamma \to G}$

${\displaystyle vol(\rho )\leq vol(M)}$

and equality only when is discreet and loyal. ${\displaystyle \rho }$

literature

• W. Goldman: Topological components of spaces of representations. Invent. Math. 93, 1988, pp. 557-607.
• G. Besson, G. Courteois, S. Gallot: Entropies et rigidités des espaces localement symétriques de courbure strictement negative. GAFA 5, 1995, pp. 731-799.
• N. Dunfield: Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds. Invent. Math. 136, 1999, pp. 623-657.
• J. Boland, C. Connell, J. Souto: Volume rigidity for finite volume manifolds. Amer. J. Math. 127, 2005, pp. 535-550.
• S. Francaviglia, B. Klaff: Maximal volume representations are Fuchsian. Geom. Dedic. 117, 2006, pp. 111-124.