Does every locally compact group $G$ have a nontrivial homomorphism into $\mathbb{R}$? Does every locally compact group (second countable and Hausdorff) topological group $G$ that is not compact have a nontrivial continuous homomorphism into $\mathbb{R}$?
Obviously for compact groups it is not possible since continuous functions send compact sets to compact sets, and there is only one (trivial) compact subgroup of $\mathbb{R}$.

Suppose that $G$ is a locally compact group with Property FH. Then, every continuous homomorphism from $G$ to $\mathbb{R}^n$ or $\mathbb{Z}^n$ is trivial. 
This follows from a corollary in the book "Kazhdan's Property T" by  Bekka, de la Harpe and Valette. 
