By W. J. Harvey, C. Maclachlan

This quantity features a collection of refereed papers awarded in honour of A.M. Macbeath, one of many best researchers within the region of discrete teams. the topic has been of a lot present curiosity of past due because it includes the interplay of a few diversified themes corresponding to workforce conception, hyperbolic geometry, and complicated research.

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Discrete Groups and Geometry

This quantity includes a choice of refereed papers offered in honour of A. M. Macbeath, one of many major researchers within the zone of discrete teams. the topic has been of a lot present curiosity of overdue because it consists of the interplay of a couple of varied subject matters resembling team conception, hyperbolic geometry, and complicated research.

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Hence for large genera we need only consider the signatures listed below: (i) ( 0 ; 2 , 2 , 2 , n ) f o r n > 3 , (ii) (0; 2, m, n) for 3 < m < n, with n > 7 if m = 3, and n > 5 if m — 4, (iii) (0; 3,ra,n) for 3 < m < n and m < 6, with n > 4 if m — 3, (iv) (0;4,4,n) for n > 4. 52 Conder and Kulkarni In cases (i) and (iv) the Riemann-Hurwitz formula gives a = 4, and similarly in the subcases (3,3,n) and (3,6,n) of case (iii) the values of a are 6 and 4 respectively. 2 precludes the possibilities (3,4, n) and (3,5, n) in case (iii), along with the subcases of (ii) in which m is odd.

2 PROPOSITION. If the sequence N8,b is admissible then b is divisible by 8. PROOF. Suppose b is an odd multiple of 4. Let G be a group in N8^ which has generators x and y of orders 2 and 4 respectively, such that xy generates a (cyclic) subgroup H of order n and index d in G. By the observations made earlier, |G| = 8g — 8 -\- Ad (where g is the genus of the associated surface), and so b = 4d — 8, implying that d is odd. 1, the core of if is a normal subgroup K of index at most d\ in G, generated by some power of xy.

Pride and R. Stohr, "The (co)homology of aspherical Coxeter groups", J. London Math. Soc. (2) 42 (1990), 49-63. [20] G. P. Scott, "Subgroups of surface groups are almost geometric", J. London Math. Soc. (2) 17 (1978), 555-565. [21] J-P. Serre, "Cohomologie des groupes discrets". In: Prospects in mathematics, Annals of Mathematics Studies 70, pp 77-169. Princeton: University Press 1971. [22] J. Tits, "Sur le groupe des automorphismes de certains groupes de Coxeter", J. Algebra 113 (1988), 346-357.

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