Although this is somewhat of an older thread, I have another suggestion for playing over the Am - Cm - Abm progression.
The challenge I see in this progression is that it lacks a definitive tonal center. It has no over-arching key that fits for all three chords. Thus, you have an important decision to make. What key shall it be in? For guidance, I would turn to the melody - assuming there is one written. For each of these minor chords, you need to decide which one will become the tonal center. My recommendation is to choose only one of the chords. That way the listener's ear will have a location in the song that creates a sense of peace and rest. Though you could have no tonal center, if that is the desire.
To create a tonal center - or key - for this progression, you will need to declare which chord will act as a vi chord, and allow the remaining minor chords to act as ii chords. This is done by purposefully choosing the 6th note for each chord. This is where you might refer back to the melody to see what notes it uses.
For example, since Am is the first chord, let's place that as the tonal center. To declare this in your soloing, include an F note - this is the minor 6th (or flat 13th) of A. Couple this with the 9th of A, which is a B. This will solidly declare to the listener that Am is the tonal center because you have forced it to act as a vi chord by choosing the F and B notes.
To reinforce this decision, over the Cm and Abm chords, play a major 6th (and 9th). The major 6th of Cm is A (and 9th is D). The major 6th of Abm is F (and the 9th of Ab is Bb).
By doing this, you tell the listener that Cm and Abm are ii chords and unresolved. They will hang in space pleasantly. The Am will root the progression by playing the F over it (as a passing tone). Since the F works double duty over the Am and Abm spaces, creating Am as the tonal center, it is your go-to soloing note. Also, an F over the Cm is an 11, which is prevalent in jazzy minor chords.
So, convert your Am into a Am7(b13). Make the Cm into a Cm11, and make the Abm into an Abminor6. Your progession becomes thus: Am7(b13) - Cm11 - Abm6.
Alternatively, you can remove all sense of key center by playing the Am also as a ii chord - playing an F# instead of an F over it.
But I would defer to the melody to help guide these decisions. In other words, start by playing the melody in your solo and detour from there.