The Type And Deformation Of Triangle Groups In Complex Hyperbolic Geometry

In this paper, we mainly investigated the type of complex hyperbolic (n1, n2,n3) triangle groups when n1 belongs to 10,11,12,13.A complex hyperbolic triangle group (n1, n2,n3) is the group of complex hyperbolic isometries generated by com-plex involutions I1,I2,I3 fixing three complex lines C1, C2, C3 such that Ck and Ck+1 meet at the angleπ/nk in complex hyperbolic space. We suppose that the deformation is starting at R-Fuchsian embedding. For each triple (n1,n2,n3), there exist one parameter family of complex hyperbolic triangle groups. We say that the triple (n1,n2,n3) has Type A if WA=I3I2I1I2 becomes elliptic before WB=I1I2I3 as the parameter changes. We say the otherwise that (n1,n2,n3) has Type B. It has been proved that the triple (n1,n2,n3) has Type A if n1<10 and Type B if n1>13. The situation is rather complicated when n1=10,11,12,13 and the type of the triple (n1,n2,n3) will be concerned with n2,n3. So, we firstly fix the number of n1 which lies in the set 10,11,12,13, then use a strait line to replace the Goldman deltoid, and the intersection point of the strait line with the deformation curve to replace the intersection point of the Goldman deltoid with the deformation curve. By comparing and verifying, we obtained that (n1,n2,n3) are Type B when n2>30,19,16,14 according to each fixed n1=10,11,12,13. Next, we have just wanted to do it for finite values of n2 one by one, which means that the number of n1 and n2 are fixed. However, there always exist infinite cases to be considered as the n3 exchanging. Therefore, we studied the monotone of the determine function about n3, under which we have gotten the type of the left triple (n1,n2,n3) by determining finite cases.We also investigated the deformation problem of the triangle group (2,4,00) in complex hyperbolic geometry. Assume that the beginning deformation was C-Fuchsian embedding, and the group was generated by three reflections about Lagrangian plane. We particularly considered a non-trivial branch in the repre-sentation space of (2,4,∞) triangle group, and obtained an interval in which the representations are discrete and faithful by constructing the fundamental domains.