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Example can be found, for instance, in Boris Weisfeiler's paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197662

His example of a discontinuous representation $\rho$ of $SO(n, {\mathbb R})$ to a semidirect product $H$ of $SO(n, {\mathbb R})$ with the abelian group ${\mathbb R}^N$ (the Lie algebra of $SO(n)$), works for $SL(2, {\mathbb R})$ as well. Actually, Weisfeiler's example is even more dramatic: The image of the compact group $SO(n)$ under $\rho$ is dense in the noncompact Lie group $H$. Weisfeiler's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups.

Example can be found, for instance, in Boris Weisfeiler's paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see

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His example of a discontinuous representation $\rho$ of $SO(n, {\mathbb R})$ to a semidirect product $H$ of $SO(n, {\mathbb R})$ with the abelian group ${\mathbb R}^N$ (the Lie algebra of $SO(n)$), works for $SL(2, {\mathbb R})$ as well. Actually, Weisfeiler's example is even more dramatic: The image of the compact group $SO(n)$ under $\rho$ is dense in the noncompact Lie group $H$. Weisfeiler's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups.

Example can be found, for instance, in Boris Weisfeller's paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197662

His example of a discontinuous representation $\rho$ of $SO(n, {\mathbb R})$ to a semidirect product $H$ of $SO(n, {\mathbb R})$ with some ${\mathbb R}^N$, works for $SL(2, {\mathbb R})$ as well. Weisfeller's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups. Actually, Weisfeller's example is even more dramatic: The image of the compact group $SO(n)$ under $\rho$ is dense in $H$.

Example can be found, for instance, in Boris Weisfeiler's paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197662

His example of a discontinuous representation $\rho$ of $SO(n, {\mathbb R})$ to a semidirect product $H$ of $SO(n, {\mathbb R})$ with the abelian group ${\mathbb R}^N$ (the Lie algebra of $SO(n)$), works for $SL(2, {\mathbb R})$ as well. Actually, Weisfeiler's example is even more dramatic: The image of the compact group $SO(n)$ under $\rho$ is dense in the noncompact Lie group $H$. Weisfeiler's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups.

Example can be found, for instance, in Boris Weisfeller's paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197662

His example of a discontinuous representation of $SO(n, {\mathbb R})$ to a semidirect product of  $SO(n, {\mathbb R})$ with some ${\mathbb R}^N$, works for $SL(2, {\mathbb R})$ as well. Weisfeller's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups.

Example can be found, for instance, in Boris Weisfeller's paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197662

His example of a discontinuous representation $\rho$ of $SO(n, {\mathbb R})$ to a semidirect product $H$ of  $SO(n, {\mathbb R})$ with some ${\mathbb R}^N$, works for $SL(2, {\mathbb R})$ as well. Weisfeller's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups. Actually, Weisfeller's example is even more dramatic: The image of the compact group $SO(n)$ under $\rho$ is dense in $H$.