By Brian H Bowditch

This quantity is meant as a self-contained creation to the elemental notions of geometric crew concept, the most principles being illustrated with a number of examples and workouts. One objective is to set up the principles of the speculation of hyperbolic teams. there's a short dialogue of classical hyperbolic geometry, for you to motivating and illustrating this.

The notes are in response to a path given by way of the writer on the Tokyo Institute of know-how, meant for fourth 12 months undergraduates and graduate scholars, and will shape the root of an identical direction in different places. Many references to extra subtle fabric are given, and the paintings concludes with a dialogue of varied parts of modern and present research.

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Define h(v) = * for each v E V and extend equivariantly to X°; then extend equivariantly skeleton by skeleton subject to the condition that for each cell o of dimension < n, h(o) lies in the closed convex hull of h(6,). Each geodesic ray e E E. based at * gives rise to a decreasing filtration (X(e,t))t,0 of X" by subcomplexes: X(e t) is defined to be the largest subcomplex of X" lying in h-' (HB(e,t)) . This filtration determines a sort of "end" of X" defined by e, since different choices of h : X" -* M lead to equivalent filtrations.

Corollary 3. Let p : G --4 PSL2(R) as in Proposition 2. Then we have: (a) p" C_ E°(G) if and only if G\ffF is compact. (b) The complement of p" fl E°(G) in p" is countable if and only if G\ has finite area. Remark. Let p : G -> PSL2(R) be as in Proposition 2, and assume G has a fundamental domain with finite area. Then p E E° (G) if and only if oo E a]HP is not a parabolic fixed point. The finite area assumption excludes the case that p(G) is "elementary", so the G-orbit of each point of alp is dense in aH' .

The author and R. Stohr have now shown that Conjecture A is true in the case p = 2 (Fixed points of automorphisms of free Lie algebras, Arch. Math. 67 (1996), pp. 281-289). Thus Conjectures B and C are also true for p = 2. Cyclic groups acting on free Lie algebras 41 Problem D. Find a basis SZ of L(2) such that SZ U -St is G(2)-invariant. An effective solution of this problem would yield a resolution of Conjecture C (and hence Conjecture A) for p = 2 because Conjecture C holds in this case if and only if no element of S2 is fixed by G(2).

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