By Jürgen Berndt

Generalized Heisenberg teams, or H-type teams, brought via A. Kaplan, and Damek-Ricci harmonic areas are relatively great Lie teams with an enormous spectrum of houses and functions. those harmonic areas are homogeneous Hadamard manifolds containing the H-type teams as horospheres.These notes include an intensive examine in their Riemannian geometry via a close remedy in their Jacobi vector fields and Jacobi operators. a few difficulties are incorporated and should expectantly stimulate additional study on those areas. The publication is written for college kids and researchers, assuming purely uncomplicated wisdom of Riemannian geometry, and it includes a short survey of the history fabric had to stick with the full therapy.

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**Example text**

Now, if we conjugate each of t0 , s0 and t0 s0 by d and take all images under the group L, we obtain three 22 The Mathieu group M24 more sets of generators for M24 , making six copies of M24 in total. As is shown later, these are the only ways in which a group isomorphic to L2 7 acting transitively on 24 letters can be extended to a copy of M24 . They are cycled by the element of order 6: zd = 1 2 15 9 13 20 3 6 4 0 5 11 7 8 21 16 14 19 10 18 12 17 22 where z is an element of order 3 commuting with L.

Since every face is joined to just one face of each tern, the sum ui i∈T 30 The Mathieu group M24 for T a tern, is the zero vector. Secondly, note that the sum of the seven uj for j joined to a fixed face, i say, is just ui , for every face other than these eight faces is joined to none or two of them. In particular, we have u = u0 + u18 + u3 + u20 + u8 + u14 + u15 ∈ B As above, we now project onto the top row of the tern array. That is to say, we define X →X∩ +u Certainly, u = 1 1 1 1 1 1 1 1 ∈ Ker ; we need to show that Im consists of the 7-dimensional space of all even subspaces of the top row.

In a Steiner system S(3,4,8), every subset of three points is contained in a unique special tetrad, so there are four dodecads of which can be added to a given octad of to give a further octad. Finally, if e is a dodecad in , then we obtain an octad by adding a dodecad of whose special tetrad of terns is contained in the set of six terns in which e lies. But, since the complement of a special tetrad is itself a special tetrad, the number of special tetrads contained in a fixed set of size 6 is equal to the number of special tetrads containing a given pair of points, namely three.