Problems in number theory and hyperbolic geometry

In the ���rst part of this thesis we generalize a theorem of Kiming and Olsson concerning the existence of Ramanujan-type congruences for a class of eta quotients. Speci���cally, we consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ��� for which their coefficients c(n) obey congruences of the form c(���n + a) ��� 0 (mod ���). We use this last result to answer a question of H.C. Chan. In the second part of this thesis [S2] we explore a natural analog of D. Calegari���s result that there are no hyperbolic once-punctured torus bundles over S^1 with trace ���eld having a real place. We prove a contrasting theorem showing the existence of several in���nite families of pairs (�����, p) such that there exist hyperbolic surface bundles over S^1 with trace ���eld of having a real place and with ���ber having p punctures and Euler characteristic ��. This supports our conjecture that with ���nitely many known exceptions there exist such examples for each pair ( �����, p).