# Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group?

Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?

• You mean, a cocompact arithmetic lattice somewhere... probably a nontrivial semisimple Lie group. – YCor Jun 23 '16 at 8:17
• @YCor Thanks. Yes. Possibly of higher rank. All examples of cocompact lattices I know, are constructed as (irreducible) lattices in semisimple groups with compact factors. So, projecting them on such a factor gives a required homomorphism. The question is equivalent to asking for an example which is not of this kind. – Jarek Kędra Jun 23 '16 at 8:45