Local rigidity

In mathematics , the concept of local rigidity describes the non-deformability of representations of groups .

A stronger concept of rigidity is the (global) Mostow rigidity .

definition

Be a matrix group , for example or , a grid and inclusion. ${\displaystyle \Gamma }$ ${\displaystyle G=GL(n,\mathbb {R} )}$ ${\displaystyle G=GL(n,\mathbb {C} )}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle \rho _{0}\colon \Gamma \to G}$

The lattice is said to be locally rigid if there is a neighborhood of in the representation variety such that all representations in this neighborhood are equivalent to (i.e. conjugated by means of an element of ). ${\displaystyle \Gamma \subset G}$ ${\displaystyle \rho _{0}}$ ${\displaystyle Hom(\Gamma ,G)}$ ${\displaystyle \rho }$ ${\displaystyle G}$

With a fixed finite generating system of , one can describe local rigidity as follows: there is a neighborhood of the neutral element in such that for every homomorphism with ${\displaystyle \Sigma }$ ${\displaystyle \Gamma }$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle \rho \colon \Gamma \to G}$

${\displaystyle \rho (\sigma )\in \sigma .U\ \ \forall \sigma \in \Sigma }$

applies: there is a with ${\displaystyle g\in G}$

${\displaystyle \rho (\gamma )=g\gamma g^{-1}\ \ \forall \gamma \in \Gamma }$.

criteria

A sufficient condition for local rigidity of a representation is the disappearance of the cohomology group , wherein the adjoint representation of designated. ${\displaystyle \rho \colon \Gamma \to G}$ ${\displaystyle H^{1}(\Gamma ,Ad\circ \rho )}$ ${\displaystyle Ad}$ ${\displaystyle G}$

From Weil-rigidity follows: A semisimple representation is exactly then locally rigid when is. ${\displaystyle H^{1}(\Gamma ,Ad\circ \rho )=0}$

Examples

Local rigidity has been proven:

• for co-compact grilles in von Selberg ${\displaystyle SL(n,\mathbb {R} ),n\geq 3}$
• for co-compact grids in by Calabi ${\displaystyle PO(n,1)=Isom(H^{n}),n\geq 3}$
• for co-compact irreducible lattices in a Lie group, not locally isometric to von Weil ${\displaystyle SL(2,\mathbb {R} )}$
• for non-co-compact lattices in Lie groups of rank 1, not locally isometric to or from Garland and Raghunathan ${\displaystyle \mathbb {R} }$ ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle SL(2,\mathbb {C} )}$
• for non-irreducible kokompakte grating in semi simple Lie groups from -Rang as a consequence of Super rigidity set of Margulis . ${\displaystyle \mathbb {R} }$ ${\displaystyle \geq 2}$

Counterexamples

Hyperbolic expansion surgery : If a non-compact hyperbolic 3-manifold is of finite volume, then an infinite number of closed hyperbolic 3-manifolds is obtained by expansion surgery . Be and the representations given by the hyperbolic structures. ${\displaystyle M}$ ${\displaystyle M_{p,q}}$ ${\displaystyle \rho \colon \pi _{1}M\to PSL(2,\mathbb {C} )}$ ${\displaystyle \rho _{p,q}\colon \pi _{1}M_{p,q}\to PSL(2,\mathbb {C} )}$

The representations can be linked to the homomorphism induced by the inclusion . The sequence of representations thus obtained converges for against , but is not too equivalent. The representation is therefore not locally rigid. ${\displaystyle \rho _{p,q}}$ ${\displaystyle M\to M_{p,q}}$ ${\displaystyle \pi _{1}M\to \pi _{1}M_{p,q}}$ ${\displaystyle \pi _{1}M\to PSL(2,\mathbb {C} )}$ ${\displaystyle p,q\to \infty }$ ${\displaystyle \rho }$ ${\displaystyle \rho }$ ${\displaystyle \rho }$

literature

• Joan Porti: Local and infinitesimal rigidity of representations of hyperbolic three manifolds. RIMS Kôkyûroku, Kyoto University Vol 1836 (2013), 154–177, online (pdf)

Individual evidence

1. ^ Atle Selberg: On discontinuous groups in higher-dimensional symmetric spaces. 1960 Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960) pp. 147-164 Tata Institute of Fundamental Research, Bombay
2. ^ Eugenio Calabi: On compact, Riemannian manifolds with constant curvature. I. 1961 Proc. Sympos. Pure Math., Vol. III pp. 155-180 American Mathematical Society, Providence, RI
3. ^ André Weil: On discrete subgroups of Lie groups. II. Ann. of Math. (2) 75 1962 578-602.
4. ^ H. Garland, MS Raghunathan: Fundamental domains for lattices in (R-) rank 1 semisimple Lie groups. Ann. of Math. (2) 92 1970 279-326.
5. ^ GA Margulis: Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1. Invent. Math. 76 (1984) no. 1, 93-120.