Download PDF by Graham A. Niblo, Martin A. Roller: Geometric group theory: - Proc. Symp. in Sussex, 1991

By Graham A. Niblo, Martin A. Roller

ISBN-10: 0521435293

ISBN-13: 9780521435291

Those volumes include survey papers given on the 1991 overseas symposium on geometric workforce conception, they usually signify the various most modern considering during this quarter. the various world's major figures during this box attended the convention, and their contributions conceal a large range of issues. quantity I includes stories of such topics as isoperimetric and isodiametric capabilities, geometric invariants of a teams, Brick's quasi-simple filtrations for teams and 3-manifolds, string rewriting, and algebraic evidence of the torus theorem, the class of teams appearing freely on R-trees, and lots more and plenty extra. quantity II is composed exclusively of a floor breaking paper via M. Gromov on finitely generated teams.

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First, we assume that 111 is not empty. Then 111 is a non empty open part of M and in each point p of Ill, we know that (Vh)p = 0. The classical Pick-Berwald theorem then implies that 111 is an open part of a nondegenerate ellipsoid or hyperboloid. Thus detS is a constant different from zero on 111" The continuity of detS then implies that fit = M. Finally, we may assume that S = 0 on the whole of M. Thus by Proposition 2, we can suppose that M is given by the equation z -- P(x,y), where P is a polynomial of degree at most k + 1, and that the canonical affine normal vector field is given by (0,0,1).

N be the Frenet frame of its directrix curve ~o" through ~p2 if and only if (i) n = 2m, (ii) ~ = ~m, (iii) ~o(z) Theorem 5 where z is a stereographic complex coordinate. - ~n(-I/~) be a linearly full minimal L e t %b : S 2 --+ CP n, n - 2m, immersion which factors through ~P~, ~o is k-point ramified for k < 2. and suppose that the directrix curve Let z be a stereographic complex coordinate such that if k ~ 0 then 4o has a higher order singularity at z - O. Then ~o is given by g/o(Z) = /kp zk' + ' " "+ kp ep (t) p-O where e_o .

Remark The condition of factoring through RP 2 is crucial here. result is not true in general. [l]). This, together with Theorem 2, is a first step in dealing with the space of minimal immersions of S ~ into Sn. Finally, combining Theorems 2 and 6, we have the following partial generalisation of Theorem 5. Theorem 7 Let # : S ~ --+ CP n be a linearly full minimal immersion which factors through Rp2. Then up to holomorphic isometries of CP n there are only finitely many minimal immersions of S 2 into CP n which factor through ~p2 and have the same singularity type as ~.

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Geometric group theory: - Proc. Symp. in Sussex, 1991 by Graham A. Niblo, Martin A. Roller

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