# Algebraic Theory of the Bianchi Groups by Fine By Fine

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Extra info for Algebraic Theory of the Bianchi Groups

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Xn}, there exists a unique closed term A* modulo eeL ~llSP such that: [ A =CCLIJIl SP A * (xn ,.. e ,DB (x ,xn ,.. 2 A-CALCULUS (x,xn ,··,xO) =CCLPllSP 1t(X ,(Xn ,··,xo». Then A • = A(A 0 1t)ld fits. Uniqueness is by ext. 0 Intuitively this result means that categorical combinatory logic is its own meta-language: in the above statement, A may be seen as the specification of a function, and A· as the code for that function. One may also say that A· "internalizes" the "function" (xn ,.. ,xo) ~ A.

XnpNDB(xo-··,xn» (MN)DB(xo-.. ,xn) = Snd 0 MDB(xo, .. 2) MDB(xO, .. ,xn) is closed. 1. Exercise Show that the following properties hold: MDB(xo-"'xn) = NDB(xo-.. y,xi+t ... xi_t,x ,xi+t,xn) =MDB(xo-.. 16), M[xof-No, .. st m+! :: {Yo,oo,Ym}, and where z is any variable. -calculus's substitution. More technically, the presence of z in the equation corresponds to the intuition that the environment is made of its useful part, and of "the rest", which z represents. Now we come to the simulation of substitution and lifting.

Xjw = A. )). We just have to notice: A =IdR,AI,DA A(App 0

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