MR0805714 Odoni, R. W. K. The Galois theory of iterates and composites of polynomials. Proc. London Math. Soc. (3) 51 (1985), no. 3, 385–414.
Might be related to Survey article on Arboreal Galois representations
For a survey on iterated monodromy groups see Nekrashevych - might be useful if you get to think about this.

MR1484879 Guralnick, Robert; Wan, Daqing Bounds for fixed point free elements in a transitive group and applications to curves over finite fields. Israel J. Math. 101 (1997), 255–287
Lemma 5.5 might be closely related to my result: it follows from Chebotarev's density theorem for function fields over finite field: given Y->X Galois cover of varieties over a finite field k, to each (zero-dimensional, degree 1) point of X we can associate "Artin symbol" --- conjugacy class in Galois group of Y/X that acts like Frobenius on the fiber. This element must act as Frobenius on algebraic closure of k as well. Besides this restriction the theorem claims that it is "equidistributed" in Gal(Y/X) (with deviation from actual equidistribution of the order of |X|/sqrt|k|.

MR1239049 Reviewed Fried, Michael D.; Guralnick, Robert; Saxl, Jan Schur covers and Carlitz's conjecture. Israel J. Math. 82 (1993), no. 1-3, 157–225.
Contains a description of monodromy groups of exceptional polynomials

MR1696200 Napolitano, Fabien Pseudo-homology of complex hypersurfaces. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 11, 1025–1030.
Arnold mentions this construction as a sort of "mixed hodge structure that remembers dimensions" --- sounds intriguing.

William Browder and Nicholas Katz have several papers on fixed points of group actions on varieties.

Cohomological methods in transformation groups
Covers Smith theory and many related ideas

Might be interesting for Chebychev polynomials question and more generally for learning to work with surfaces

Anabelian intersection theory - explains some birational invariants of incomplete parshin chains. Might be useful for my work on essential dimension (of p-groups?).