By Roger C. Lyndon

ISBN-10: 0521316944

ISBN-13: 9780521316941

This e-book, which used to be initially released in 1985 and has been translated and revised by means of the writer from notes of a direction, is an creation to definite relevant principles in staff conception and geometry. Professor Lyndon emphasises and exploits the well known connections among the 2 matters and, when holding the presentation at a degree that assumes just a easy historical past in arithmetic, leads the reader to the frontiers of present study on the time of book. The remedy is concrete and combinatorial with a minimum use of analytic geometry. within the curiosity of the reader's instinct, many of the geometry thought of is two-dimensional and there's an emphasis on examples, either within the textual content and within the difficulties on the finish of every bankruptcy.

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The proof reduces to the case that p is a simple loop (does not intersect itself), where, inductively, by running around the other side of some region A , one can reduce the number of regions A enclosed by p. D For the tessellation F of type (3,6), by equilateral triangular regions, six at each vertex, the argument differs from the case above only in that the interior angles of A are different. We find a presentation G = < a,3,Y : a 2 - 3 2 = Y 2 = ( 3 Y ) 3 = (Ya) 2 = (a3) 6 = 1 >. The remaining tessellation, of type (6,3) is dual to one of type (3,6), and hence has the same symmetry group G; this can also be deduced from the fact that a single fundamental region A serves for both of them.

1, b = a , Thus we can delete 2 3 the first three relators to obtain a presentation < a,b,c: b , c , bcb c , a be >. Finally, by a step of type 2* we can delete the generator a together with the relator a < b,c: b , c , bcb REMARK. c be, to obtain >, as desired. One should be warned that this theorem does not provide an algorithm for deciding whether two given finite presentations define isomorphic groups. This follows from the fact that the word problem for groups is unsolvable: there exists no algorithm which, given any free group F with a finite basis X, a finite subset R of F, and an element r of F, decides whether r belongs to the normal closure of R in F.

P^ = pp~. If P-jPo = PPo i-s a translation T , then the axes of all of Pi> P9> Po> P m ust be perpendicular to T , and hence parallel. p,P9 = pp through 0. If is a rotation with center 0, then all the axes must pass • THEOREM. (1) The product of two translations is a translation. (2) The product of two rotations is a rotation except if the sum of their angles is 0 modulo 2TT, in which case it is a translation. (3) The product of a translation and a nontrivial rotation is a rotation. (4) The product of a nontrivial translation and a reflection is a glide reflection except if the axis of the reflection is perpendicular to 27 the translation, in which case it is a reflection.

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