TITLE: If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?
QUESTION [7 upvotes]: We know that if $G$ is a topological group that contains a torsion element and $G$ satisfies additional conditions such as $G$ discrete or $G$ finite-dimensional, then the classifying space $BG$ is infinite-dimensional.
My question: can we remove the additional condition such that the statement still holds? Namely, is there any counterexample for $G$ infinite-dimensional?
Edit: Sorry, G contains a torsion element should be G is not torsion-free. Thank you for your thoughtful inputs.

REPLY [13 votes]: Here is sort of a canonical example.
Consider $GL(\mathbb H)$ the group of invertible operators on a Hilbert space. By Kuipers theorem it is contractible. But $GL(\mathbb H)$ acts freely and properly on itself. Hence the classifying space is $BGL(\mathbb H)=GL(\mathbb H)/GL(\mathbb H)$ a point. Consequences are that Hilbert bundles are trivial over paracompact spaces, K-theory is represented by Fredholm operators etc.
Of course $GL(\mathbb H)$ contains a lot of torsion. Any finite group is a subgroup of $GL(\mathbb H)$ for example.