CP(1) This is the conjugate As we defined it, S cCP(2)~ /=o/ and the restriction /~"/S > CP(1) may be readily identified with - the quotient map /2: S 52 - Thus we have constructed > S/T. an n-fold ramified covering of CP(1) which branches at . 9cal uniformizer~ At each fixed point ~ z~ with respect t__oowhich T(z) = A z . We choose a fixed point Ely - ~ J holomorphic o]. (z)~ defined for Izl < i~ such that 3 a) Rj(O) = b) (Rj(~))n = i + ~, for I~I < 1. ,~ The required uniformizer ~(z) (T,S) there is a is = [t, -Rj(zn),z] Note that I qRj(zn)) n + z n = 1 - 1 - z n + zn = O and We hope t o compute t r a c e (T*I hOwl(s)) finite fixed point set.

We call such an action ( ~ ' ) - f r e e . bordism group (H; Sf for such pairs. We define We say that (B n, H) bounds if and only if there is (Wn+l which is ( ~ ~ ) - f r e e (i) (S n, H) C H) and ( ~ W n+l, H) as a compact, regular oriented submanifold (2) if x e ~ w n + l k B n then H x is conjugate to an element o f ~ . Of course (BI, H ) ~ ( B , union (B~, U -B~, H) if and only if the disjoint H) bounds. The resulting set of equivalence classes is denoted by ~ ( H ; ~ ~ ) n and is - 34 - made into a group by using disjoint union to define addition.

### Asymptotic invariants of infinite groups by Gromov M.

by James

4.4