Local rigidity

Let ${\displaystyle \Gamma }$ be a group generated by a finite number of elements ${\displaystyle g_{1},\ldots ,g_{n}}$ and ${\displaystyle G}$ a Lie group. Then the map ${\displaystyle \mathrm {Hom} (\Gamma ,G)\to G^{n}}$ defined by ${\displaystyle \rho \mapsto (\rho (g_{1}),\ldots ,\rho (g_{n}))}$ is injective and this endows ${\displaystyle \mathrm {Hom} (\Gamma ,G)}$ with a topology induced by that of ${\displaystyle G^{n}}$. If ${\displaystyle \Gamma }$ is a subgroup of ${\displaystyle G}$ then a deformation of ${\displaystyle \Gamma }$ is any element in ${\displaystyle \mathrm {Hom} (\Gamma ,G)}$. Two representations ${\displaystyle \phi ,\psi }$ are said to be conjugated if there exists a ${\displaystyle g\in G}$ such that ${\displaystyle \phi (\gamma )=g\psi (\gamma )g^{-1}}$ for all ${\displaystyle \gamma \in \Gamma }$. See also character variety.

The simplest statement is when ${\displaystyle \Gamma }$ is a lattice in a simple Lie group ${\displaystyle g}$ and the latter is not locally isomorphic to ${\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}$ or ${\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}$ and ${\displaystyle \Gamma }$ (this means that its Lie algebra is not that of one of these two groups).

There exists a neighbourhood ${\displaystyle U}$ in ${\displaystyle \mathrm {Hom} (\Gamma ,G)}$ of the inclusion ${\displaystyle i:\Gamma \subset G}$ such that any ${\displaystyle \phi \in U}$ is conjugated to ${\displaystyle i}$.

Whenever such a statement holds for a pair ${\displaystyle G\supset \Gamma }$ we will say that local rigidity holds.

Local rigidity holds for cocompact lattices in ${\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}$. A lattice ${\displaystyle \Gamma }$ in ${\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}$ which is not cocompact has nontrivial deformations coming from Thurston's hyperbolic Dehn surgery theory. However, if one adds the restriction that a representation must send parabolic elements in ${\displaystyle \Gamma }$ to parabolic elements then local rigidity holds.

In this case local rigidity never holds. For cocompact lattices a small deformation remains a cocompact lattice but it may not be conjugated to the original one (see Teichmüller space for more detail). Non-cocompact lattices are virtually free and hence have non-lattice deformations.

Local rigidity holds for lattices in semisimple Lie groups providing the latter have no factor of type A1 (i.e. locally isomorphic to ${\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}$ or ${\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}$) or the former is irreducible.