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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The sensible analytic houses of Weyl transforms as bounded linear operators on $ L^{2}({\Bbb R}^{n}) $ are studied when it comes to the symbols of the transforms. The boundedness, the compactness, the spectrum and the practical calculus of the Weyl remodel are proved intimately. New effects and methods at the boundedness and compactness of the Weyl transforms by way of the symbols in $ L^{r}({\Bbb R}^{2n}) $ and when it comes to the Wigner transforms of Hermite features are given.

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.

**Transformations of Manifolds and Application to Differential Equations**

The interplay among differential geometry and partial differential equations has been studied because the final century. This courting is predicated at the incontrovertible fact that lots of the neighborhood houses of manifolds are expressed by way of partial differential equations. The correspondence among convinced periods of manifolds and the linked differential equations could be helpful in methods.

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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.