By F. Thomas Farrell and L. Edwin Jones

Aspherical manifolds--those whose common covers are contractible--arise classically in lots of components of arithmetic. They ensue in Lie crew concept as yes double coset areas and in man made geometry because the house varieties keeping the geometry. This quantity includes lectures introduced via the 1st writer at an NSF-CBMS local convention on K-Theory and Dynamics, held in Gainesville, Florida in January, 1989. The lectures have been basically concerned about the matter of topologically characterizing classical aspherical manifolds. This challenge has for the main half been solved, however the three- and four-dimensional situations stay an important open questions; Poincare's conjecture is heavily regarding the three-d challenge. one of many major effects is closed aspherical manifold (of size $\neq$ three or four) is a hyperbolic area if and provided that its primary workforce is isomorphic to a discrete, cocompact subgroup of the Lie staff $O(n,1;{\mathbb R})$. one of many book's subject matters is how the dynamics of the geodesic circulation should be mixed with topological keep an eye on thought to check competently discontinuous team activities on $R^n$. the various extra technical themes of the lectures were deleted, and a few extra effects received because the convention are mentioned in an epilogue. The booklet calls for a few familiarity with the fabric contained in a uncomplicated, graduate-level path in algebraic and differential topology, in addition to a few basic differential geometry.

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**Example text**

20) The kernel of the free term in this equation has a weak singularity on the diagonal, and so is not a componentwise continuous function on F X F. 20), it is natural to iterate the equation once. We introduce the operator The equation for SS \ ( s ) = SS ( s) - � ( K ) . 22) is regarded as a Fredholm integral equation for the kernel s; 1 ) of the operator \(s), which is regarded as an unknown function of the variable The other variables are regarded as parameters. 22) are operators in the space V, it is natural to consider the following system of h equations (h = dim V): SS z, z' z.

Kfj_ 1 + k ( /3 ) . \ k a • We shall let 6 ( s ) denote this matrix. 21 ) 2 ) 6 (s) 6 { I - s ) = Ik ' where Ik is the identity matrix of order k = kef; X ), and * is the adjoint matrix. 5 it is not hard to prove the following theorem giving a functional equation for ",,( z ; s; 0'; I). Here we shall limit ourselves to the statement of the theorem. 3 . 6. THEOREM t/; ( z ; 1 - s ; a ; I ) = where 1 :;;;; a :;;;; n and 1 :;;;; n kfJ @5 al , fJ ( 1 - s ) t/;(z ; s; /3 ; k ) , � � k fJ= i k= i I :;;;; k a .

According to the remark following Theorem 1 . 7, for y E r we have c( g� I ygp ) =1= 0, as long as a =1= {3. Furthermore, if a = {3, then c( g�l yga) = 0, if and only if y E ra' a, {3 = I , . . , n o In ( 3 . 2) we shall first suppose that a =1= {3. We have aj { y , s ) = 2: PpX * ( YSpm ) e,{ a ) l lys( g� l yS;lgpz ) exp { - 2 7TijX ) dx , 2: 0 y E f,,\ f/fp m E Z ( 3 . 4 ) where y runs through the set of double cosets raY rp . I t is not hard to verify the equalities Sfgp = gp S::;. Pp X * ( ySpm ) = Pp X*{ y ) , U sing them in the right side of ( 3 .