Let $A,B$ be finitely generated groups with a common **finite** subgroup $C$. Suppose that $[A : C] > 2$.  

> Must $A *_C B$ have positive rank gradient?

See http://mathoverflow.net/questions/223715/which-3-manifolds-have-positive-rank-gradient/223737#223737 for a definiton of rank gradient.

The assumption on the index is necessary (otherwise we take $A,B = \mathbb{Z}/2\mathbb{Z}, C = \{1\}$). 

I think that there should be a proof using Bass-Serre theory.