By P. Deligne (auth.), Roger Howe (eds.)
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Extra info for Discrete Groups in Geometry and Analysis: Papers in Honor of G.D. Mostow on His Sixtieth Birthday
Our main Also first author was partially supported by NSF grant #MCS77-24l03, the second by NSF grant #MCS-8200639. 49 H(M x R) is an interesting space closely related to C(M). Dur first main result is a lower bound for the dimensions of the three previous deformation spaces by r, the largest number of disjoint, non-singular, totally geodesic hypersurfaces contained in surface of genus g then r = 3g - 3. M. If M is a hyperbolic From this bound, it is easily shown that the deformation spaces have arbitrarily large dimension as M varies.
NI ,nI) on the set of its 42 vertices. (~,vl)' - This function can be described by the characteristic pairs ••• , K[[xl ,x 2 ]] (~ g ,v) of g C, which depend only on the factor ring of by the principal ideal generated by least recall that ~i' vi f(x); cf. . Vi ::: 2 for all i, ~/vl > 1, and ~i/vi - ~i-l > 0 and that a(f), defined locally, is given by Now if EI is a nonbranching vertex and if neighboring vertices with We at are relatively prime integers satisfying EI" created after EI EI" for EI" by P - 1 i > 1 are its quadratic transformations, then On the other hand if EI is a branching vertex and if are its neighboring vertices with EI'" created after EI" EI EI'" by EI'" P - 1 quadratic transformations, then Furthermore, at least in the p-adic case, the first relation is responsible for the fact that Z~(w) EI has no contribution to the poles of and the second relation is responsible for the fact that does have a contribution to the poles of EI Z~(ws).
It seems the first resu1t showing the non-trivia1ity of Hom(f,SO(n+1,1» for n ~ 3 was Apanasov . The matter was great1y c1arified by Thurston's idea of bending a Fuchsian group, see Su11ivan  or Kourouniotis . There are a number of technica1 theorems contained in this paper in addition to the main resu1ts a11uded to above. For the reader's convenience we brief1y state them in order of occurrence. Section 1 defines stab1e representations, characterizes them in terms of parabo1ic subgroups and proves they are Zariski open in Hom(f,G).
Discrete Groups in Geometry and Analysis: Papers in Honor of G.D. Mostow on His Sixtieth Birthday by P. Deligne (auth.), Roger Howe (eds.)
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