Local rigidity

<h2 id="history">History</h2>
<p>The first such theorem was proven by <a href="/facts/Atle_Selberg/3JDSUN2k">Atle Selberg</a> for co-compact discrete subgroups of the unimodular groups 
  
    
      
        
          
            S
            L
          
          
            n
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle \mathrm {SL} _{n}(\mathbb {R} )}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> Shortly afterwards a similar statement was proven by <a href="/facts/Eugenio_Calabi/jm1DIMJm">Eugenio Calabi</a> in the setting of fundamental groups of compact hyperbolic manifolds. Finally, the theorem was extended to all co-compact subgroups of semisimple Lie groups by <a href="/facts/Andr%25C3%25A9_Weil/Q4Dd0Gr4">André Weil</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> The extension to non-cocompact lattices was made later by Howard Garland and <a href="/facts/M._S._Raghunathan/U9RRz5Cr">Madabusi Santanam Raghunathan</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
The result is now sometimes referred to as Calabi—Weil (or just Weil) rigidity.
</p>
<h2 id="statement">Statement</h2>
<h3>Deformations of subgroups</h3>
<p>Let 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 be a group <a href="/facts/Generating_set_of_a_group/B9q5lnFY">generated</a> by a finite number of elements 
  
    
      
        
          g
          
            1
          
        
        ,
        …
        ,
        
          g
          
            n
          
        
      
    
    {\displaystyle g_{1},\ldots ,g_{n}}
  
 and 
  
    
      
        G
      
    
    {\displaystyle G}
  
 a Lie group. Then the map 
  
    
      
        
          H
          o
          m
        
        (
        Γ
        ,
        G
        )
        →
        
          G
          
            n
          
        
      
    
    {\displaystyle \mathrm {Hom} (\Gamma ,G)\to G^{n}}
  
 defined by 
  
    
      
        ρ
        ↦
        (
        ρ
        (
        
          g
          
            1
          
        
        )
        ,
        …
        ,
        ρ
        (
        
          g
          
            n
          
        
        )
        )
      
    
    {\displaystyle \rho \mapsto (\rho (g_{1}),\ldots ,\rho (g_{n}))}
  
 is injective and this endows 
  
    
      
        
          H
          o
          m
        
        (
        Γ
        ,
        G
        )
      
    
    {\displaystyle \mathrm {Hom} (\Gamma ,G)}
  
 with a topology <a href="/facts/Induced_topology/xH9J6U04">induced</a> by that of 
  
    
      
        
          G
          
            n
          
        
      
    
    {\displaystyle G^{n}}
  
. If 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 is a subgroup of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 then a <i>deformation</i> of 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 is any element in 
  
    
      
        
          H
          o
          m
        
        (
        Γ
        ,
        G
        )
      
    
    {\displaystyle \mathrm {Hom} (\Gamma ,G)}
  
. Two representations 
  
    
      
        ϕ
        ,
        ψ
      
    
    {\displaystyle \phi ,\psi }
  
 are said to be conjugated if there exists a 
  
    
      
        g
        ∈
        G
      
    
    {\displaystyle g\in G}
  
 such that 
  
    
      
        ϕ
        (
        γ
        )
        =
        g
        ψ
        (
        γ
        )
        
          g
          
            −
            1
          
        
      
    
    {\displaystyle \phi (\gamma )=g\psi (\gamma )g^{-1}}
  
 for all 
  
    
      
        γ
        ∈
        Γ
      
    
    {\displaystyle \gamma \in \Gamma }
  
. See also <a href="/facts/Character_variety/XIRJ6koa">character variety</a>.
</p>
<h3>Lattices in simple groups not of type A1 or A1 × A1</h3>
<p>The simplest statement is when 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 is a lattice in a simple Lie group 
  
    
      
        G
      
    
    {\displaystyle G}
  
 and the latter is not locally isomorphic to 
  
    
      
        
          
            S
            L
          
          
            2
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}
  
 or 
  
    
      
        
          
            S
            L
          
          
            2
          
        
        (
        
          C
        
        )
      
    
    {\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}
  
 and 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 (this means that its Lie algebra is not that of one of these two groups).
</p>
<i>There exists a neighbourhood 
  
    
      
        U
      
    
    {\displaystyle U}
  
 in 
  
    
      
        
          H
          o
          m
        
        (
        Γ
        ,
        G
        )
      
    
    {\displaystyle \mathrm {Hom} (\Gamma ,G)}
  
 of the inclusion 
  
    
      
        i
        :
        Γ
        ⊂
        G
      
    
    {\displaystyle i:\Gamma \subset G}
  
 such that any 
  
    
      
        ϕ
        ∈
        U
      
    
    {\displaystyle \phi \in U}
  
 is conjugated to 
  
    
      
        i
      
    
    {\displaystyle i}
  
. </i>
<p>Whenever such a statement holds for a pair 
  
    
      
        G
        ⊃
        Γ
      
    
    {\displaystyle G\supset \Gamma }
  
 we will say that local rigidity holds.
</p>
<h3>Lattices in SL(2,C)</h3>
<p>Local rigidity holds for cocompact lattices in 
  
    
      
        
          
            S
            L
          
          
            2
          
        
        (
        
          C
        
        )
      
    
    {\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}
  
. A lattice 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 in 
  
    
      
        
          
            S
            L
          
          
            2
          
        
        (
        
          C
        
        )
      
    
    {\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}
  
 which is not cocompact has nontrivial deformations coming from Thurston's <a href="/facts/Hyperbolic_Dehn_surgery/Eus4wJkF">hyperbolic Dehn surgery</a> theory. However, if one adds the restriction that a representation must send parabolic elements in 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 to parabolic elements then local rigidity holds.
</p>
<h3>Lattices in SL(2,R)</h3>
<p>In this case local rigidity never holds (except <a href="/facts/Cocompact_group_action/d9IoKywY"> cocompact</a> <a href="/facts/Triangle_groups/LrEs3xVw"> triangle groups</a>). For cocompact lattices a small deformation remains a cocompact lattice but it may not be conjugated to the original one (see <a href="/facts/Teichm%25C3%25BCller_space/IjwtcRyb">Teichmüller space</a> for more detail). Non-cocompact lattices are virtually free and hence have non-lattice deformations.
</p>
<h3>Semisimple Lie groups</h3>
<p>Local rigidity holds for lattices in <a href="/facts/Simple_Lie_group/LkHTNYU9"> semisimple Lie groups</a> providing the latter have no factor of type A1 (i.e. locally isomorphic to 
  
    
      
        
          
            S
            L
          
          
            2
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}
  
 or 
  
    
      
        
          
            S
            L
          
          
            2
          
        
        (
        
          C
        
        )
      
    
    {\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}
  
) or the former is irreducible.
</p>
<h2 id="other-results">Other results</h2>
<p>There are also local rigidity results where the ambient group is changed, even in case where superrigidity fails. For example, if 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 is a lattice in the <a href="/facts/Unitary_group/lEvudIhU">unitary group</a> 
  
    
      
        
          S
          U
        
        (
        n
        ,
        1
        )
      
    
    {\displaystyle \mathrm {SU} (n,1)}
  
 and 
  
    
      
        n
        ≥
        2
      
    
    {\displaystyle n\geq 2}
  
 then the inclusion 
  
    
      
        Γ
        ⊂
        
          S
          U
        
        (
        n
        ,
        1
        )
        ⊂
        
          S
          U
        
        (
        n
        +
        1
        ,
        1
        )
      
    
    {\displaystyle \Gamma \subset \mathrm {SU} (n,1)\subset \mathrm {SU} (n+1,1)}
  
 is locally rigid.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p><p>A uniform lattice 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 in any compactly generated topological group 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is <i>topologically locally rigid</i>, in the sense that any sufficiently small deformation 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 of the inclusion 
  
    
      
        i
        :
        Γ
        ⊂
        G
      
    
    {\displaystyle i:\Gamma \subset G}
  
 is injective and 
  
    
      
        φ
        (
        Γ
        )
      
    
    {\displaystyle \varphi (\Gamma )}
  
 is a uniform lattice in 
  
    
      
        G
      
    
    {\displaystyle G}
  
. An irreducible uniform lattice in the isometry group of any proper geodesically complete 
  
    
      
        
          C
          A
          T
        
        (
        0
        )
      
    
    {\displaystyle \mathrm {CAT} (0)}
  
-space  not isometric to the hyperbolic plane and without Euclidean factors is locally rigid.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="proofs-of-the-theorem">Proofs of the theorem</h2>
<p>Weil's original proof is by relating deformations of a subgroup 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 in 
  
    
      
        G
      
    
    {\displaystyle G}
  
 to the first <a href="/facts/Group_cohomology/ncLTIHj4">cohomology</a> group of 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 with coefficients in the Lie algebra of 
  
    
      
        G
      
    
    {\displaystyle G}
  
, and then showing that this cohomology vanishes for cocompact lattices when 
  
    
      
        G
      
    
    {\displaystyle G}
  
 has no simple factor of absolute type A1. A more geometric proof which also work in the non-compact cases uses <a href="/facts/Charles_Ehresmann/KNtePSlk">Charles Ehresmann</a> (and <a href="/facts/William_Thurston/K1Qe8801">William Thurston</a>'s) theory of 
  
    
      
        (
        G
        ,
        X
        )
      
    
    {\displaystyle (G,X)}
  
 structures.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Selberg, Atle (1960). "On discontinuous groups in higher-dimensional symmetric spaces". Contributions to functional theory. Tata Institut, Bombay. pp. 100–110. <a href="/wiki/Atle_Selberg" target="_blank">/wiki/Atle_Selberg</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Weil, André (1960), "On discrete subgroups of Lie groups", Annals of Mathematics, Second Series, 72 (2): 369–384, doi:10.2307/1970140, ISSN 0003-486X, JSTOR 1970140, MR 0137792 <a href="/wiki/Andr%C3%A9_Weil" target="_blank">/wiki/Andr%C3%A9_Weil</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Weil, André (1962), "On discrete subgroups of Lie groups. II", Annals of Mathematics, Second Series, 75 (3): 578–602, doi:10.2307/1970212, ISSN 0003-486X, JSTOR 1970212, MR 0137793 <a href="/wiki/Andr%C3%A9_Weil" target="_blank">/wiki/Andr%C3%A9_Weil</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Garland, Howard; Raghunathan, M.~S. (1970). "Fundamental domains for lattices in R-rank 1 Lie groups". Annals of Mathematics. 92: 279–326. doi:10.2307/1970838. JSTOR 1970838. <a href="/wiki/Howard_Garland" target="_blank">/wiki/Howard_Garland</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Goldman, William; Millson, John (1987), "Local rigidity of discrete groups acting on complex hyperbolic space", Inventiones Mathematicae, 88 (3): 495–520, Bibcode:1987InMat..88..495G, doi:10.1007/bf01391829, S2CID 15347622 <a href="/wiki/William_Goldman" target="_blank">/wiki/William_Goldman</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Gelander, Tsachik; Levit, Arie (2017), "Local rigidity of uniform lattices", Commentarii Mathematici Helvetici, arXiv:1605.01693 <a href="/wiki/Tsachik_Gelander" target="_blank">/wiki/Tsachik_Gelander</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Bergeron, Nicolas; Gelander, Tsachik (2004). "A note on local rigidity". Geometriae Dedicata. 107. Kluwer: 111–131. arXiv:1702.00342. doi:10.1023/b:geom.0000049122.75284.06. S2CID 54064202. <a href="/wiki/Nicolas_Bergeron" target="_blank">/wiki/Nicolas_Bergeron</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Local rigidity open-in-new

Local rigidity