When considering homomorphisms from a fixed group H to a (torsion-free) hyperbolic group G, natural questions
arise: When are there infinitely many homomorphisms from G to H? Infinitely many up to conjugacy? This leads to the study
of which finitely generated subgroups of G split nontrivially over {1} or over Z. I will describe some of this story, and present
some algorithmic results (in both positive and negative directions). This is joint work with Henry Wilton.