TITLE: Does each discrete solvable group admit an injective homomorphism to a compact topological group?
QUESTION [8 upvotes]: It is well-known that each abelian group admits an injective homomorphism to some compact topological group (for example to its Bohr compactification). Is the same fact true for solvable groups?
Question 1. Does every solvable group admit an injective homomorphism to a compact Hausdorff topological group?
Equivalently:
Question 2. Is the Bohr topology of every discrete solvable group Hausdorff? 
The Bohr topology is the largest totally bounded group topology on the group.

REPLY [2 votes]: According to Proposition 3.3 from Dikranjan and Toller [Topology and its Applications 159 (2012) 2951-2972], the Heisenberg group H_K over an infinite field K of characteritic 0 is not maximally almost periodic. This provides
a negative answer to the question even for nilpotent groups of class 2.