# Implications of a local isomorphism on discrete subgroups and Kazhdan property (T)

In the book "Discrete groups, expanding graphs and invariant measures" by Alexander Lubotzky, page 37, the author says that all finitely generated discrete Kazhdan subgroups of $$SO(3)$$ are finite, and the same follows for $$SO(4)$$, since $$SO(4)$$ is locally isomorphic to $$SO(3) \times SO(3)$$.

I know that $$SO(4)$$ is locally isomorphic to $$SO(3) \times SO(3)$$ but I really can't see why this would imply that a finitely generated discrete Kazhdan subgroup of $$SO(4)$$ must be finite. Indeed, let $$U \subset SO(4)$$ be the domain of the local isomorphism, and let $$\Gamma \leq SO(4)$$ be a finitely generated discrete Kazhdan subgroup. First of, I don't see how the local isomorphism can relate $$\Gamma$$ to a Kazhdan subgroup of $$SO(3) \times SO(3)$$, and even with that, I don't see why the fact that $$\Gamma \cap U$$ is finite should imply that $$\Gamma$$ is finite. I mean, $$\Gamma$$ is discrete, so if we take $$U$$ small enough $$\Gamma \cap U$$ is a singleton, which gives no new information...

I assume that you take for granted that no infinite subgroup of $$\mathrm{SO}(3)$$ has Property T when endowed with the discrete topology.
Suppose by contradiction that $$\mathrm{SO}(4)$$ has an infinite subgroup $$\Gamma$$ having Property T (when endowed with the discrete topology). Then the image $$\bar{\Gamma}$$ of $$\Gamma$$ in $$\mathrm{PSO}(4)=\mathrm{SO}(4)/\{\pm I_4\}$$ is still infinite, and still has Property T. Then $$\mathrm{PSO}(4)$$ is isomorphic to $$\mathrm{SO}(3)^2$$, and it follows that the image of $$\bar{\Gamma}$$ in at least one of the two $$\mathrm{SO}(3)$$ factors is infinite and this is a contradiction.
• Ok so indeed it does not follow from the local isomorphism of $SO(4)$ with $SO(3)^2$, but from the stronger fact that $PSO(4) \cong SO(3)^2$. How does one prove that fact? I deduced the local isomorphism from the isomorphism of Lie algebras, I think one needs something more explicit here – frafour Nov 29 '18 at 22:53