It is well-known that each abelian group admits an injective homomorphism to some compact topological group (for example to its [Bohr compactification](https://en.wikipedia.org/wiki/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.



Does each discrete solvable group admit an injective homomorphism to a compact topological group?
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. 


