By Ken'ichi Ohshika

This publication offers with geometric and topological elements of discrete teams. the most themes are hyperbolic teams as a result of Gromov, computerized workforce idea, invented and built by way of Epstein, whose topics are teams that may be manipulated via desktops, and Kleinian staff idea, which enjoys the longest culture and the richest contents in the idea of discrete subgroups of Lie teams. what's universal between those 3 periods of teams is that once noticeable as geometric items, they've got the houses of a negatively curved house instead of a definitely curved house. As Kleinian teams are teams performing on a hyperbolic area of continuous unfavorable curvature, the approach hired to check them is that of hyperbolic manifolds, standard examples of negatively curved manifolds. even supposing hyperbolic teams within the feel of Gromov are even more normal gadgets than Kleinian teams, you possibly can follow for them arguments and methods which are really just like these used for Kleinian teams. computerized teams are extra normal items, together with teams having homes of areas of curvature zero. nonetheless, relationships among automated teams and hyperbolic teams are tested right here utilizing rules encouraged by means of the learn of hyperbolic manifolds. In all of those 3 subject matters, there's a ``soul'' of unfavourable curvature upholding the speculation. the quantity might make a good textbook for a graduate-level path in discrete teams.

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Using the Jordan canonical form, show that every n × n matrix A can be written as A = S + N , with S diagonalizable (over C), N nilpotent, and SN = N S. Recall that the Jordan canonical form is block diagonal, with 10. EXERCISES each block of the form ∗ λ .. 0 . 51 . λ 3. Let X and Y be n × n matrices. Show that there exists a constant C such that C e(X+Y )/m − eX/m eY /m ≤ 2 m for all integers m ≥ 1. 4. Using the Jordan canonical form, show that every n × n complex matrix A is the limit of a sequence of diagonalizable matrices.

3, we have λ e 1 0 −1 .. eX = C C . λn 0 e D 30 3. LIE ALGEBRAS AND THE EXPONENTIAL MAPPING Thus if you can explicitly diagonalize X, you can explicitly compute eX . 1) is real. For example, take −a 0 0 a X= 1 and i respectively. Thus the invertible matrix i 1 Then the eigenvectors of X are 1 i C= . , with eigenvalues −ia and ia, i 1 1 0 and to the eigenvectors of X, and so (check) 0 1 C −1 XC is a diagonal matrix D. Thus X = CDC −1 : maps the basis vectors eX = 1 i i 1 = cos a sin a e−ia 0 − sin a cos a 0 eia 1/2 −i/2 −i/2 1/2 .

Integrating term-by-term and noting that log 1 = 0 gives log(1 − x) = − x + x2 2 + x3 3 + (1−z)3 3 +··· . Taking z = 1 − x (so that x = 1 − z), we have log z = − (1 − z) + (1−z)2 2 ∞ (−1)m+1 = m=1 +··· (z − 1)m . m This series has radius of convergence one, and defines a complex analytic function on the set {|z − 1| < 1}, which coincides with the usual logarithm for real z in the interval (0, 2). Now, exp(log z) = z for z ∈ (0, 2), and by analyticity this identity continues to hold on the whole set {|z − 1| < 1}.