By Brian H Bowditch

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

The notes are in response to a direction given via the writer on the Tokyo Institute of know-how, meant for fourth 12 months undergraduates and graduate scholars, and will shape the root of the same direction somewhere else. Many references to extra refined fabric are given, and the paintings concludes with a dialogue of assorted parts of modern and present research.

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1±_ G is a discontinuous subgroup of E and T = G n T is trivial, then G is a finite group, either cyclic or dihedral. Proof. We have seen that if G contains no nontrivial translation, then it fixes some point 0, that is, G = G n , whence the preceding theorem applies. • THEOREM. _If. G is a discontinuous subgroup of E and T = G O T is infinite cyclic, then G is a frieze group. Proof. This was the definition of a frieze group. • The case remains that G is a discontinuous subgroup of E and T = G n T is free abelian of rank 2.

We show that the cyclic group C, is the direct product C^ of cyclic groups Q of order 2 and C the two presentations < a: a define isomorphic groups. of order 3, that is, that = 1 > and < b , c : b =1, c = 1, be = cb > Of course, this could be done directly, but here we give a proof in terms of Tietze transformations. To conform to the usage above, we rewrite the presentations in the forms < a: a > and < b,c: b , c , beb ft presentation < a: a c ^ >. -13 "I 2 v T , of type 2 we pass to the presentation % a,b,c: a , b a , c a >.

2 THEOREM. There are exactly seven geometrically distinct types of frieze groups, with presentations as given above. They fall into the four isomorphism types of C^, D^, C^ x C ? , D^ * C . In the following figure we show friezes with symmetry groups of the seven types. In these figures, x is a horizontal translation. lines indicate axes of reflections or glide reflections. Broken In the figures 1 2 for F , F , and F_, small circles mark centers of rotational symmetry. r Q -X F> r r f r T x HE'"' ±+\ 43 3.

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