The first example of a Lie group is the general linear group GL(n,R) = {A ∈ Matn(R)| det(A) 6= 0} of invertible n × n matrices. It is an open subset of Matn(R), hence a submanifold, and the smoothness of group multiplication follows since the product map for Matn(R) is obviously smooth. Our next example is the orthogonal group O(n) = {A ∈ Matn(R)| AA = I}. To see that it is a Lie group, it suffices to show that O(n) is an embedded submanifold of Matn(R). In order to construct submanifold charts, we use the exponential map of matrices

Lie Groups and Lie Algebras

In this chapter, we describe nine families of Moufang polygons. In Part III of this book, we prove that every Moufang polygon is isomorphic to one of the Moufang polygons described in this chapter.

An Inventory of Moufang Polygons

Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.

Homological stability for classical groups

We study lattices on ~A_2 buildings that preserve types, act regularly on each type of edge, and whose vertex stabilizers are cyclic. We show that several of their properties, such as their automorphism group and isomorphism class, can be determined from purely combinatorial data. As a consequence we can show that the number of such lattices (up to isomorphism) grows super-exponentially with the thickness parameter q. 
We look in more detail at the 3295 lattices with q in {2,3,4,5}. We show that with one exception for each q these are all exotic. For the exotic examples we prove that the automorphism group of the lattice and of the building coincide, and that two lattices are quasi-isometric only if they are isomorphic.

On panel-regular ~A_2 lattices

Let K be the field of formal Laurent series over the finite field of order q. We construct cocompact lattices \Gamma'_0 < \Gamma_0 in the group G = PGL_d(K) which are type-preserving and act transitively on the set of vertices of each type in the building associated to G. The stabiliser of each vertex in \Gamma'_0 is a Singer cycle and the stabiliser of each vertex in \Gamma_0 is isomorphic to the normaliser of a Singer cycle in PGL_d(q). We then show that the intersections of \Gamma'_0 and \Gamma_0 with PSL_d(K) are lattices in PSL_d(K), and identify the pairs (d,q) such that the entire lattice \Gamma'_0 or \Gamma_0 is contained in PSL_d(K). Finally we discuss minimality of covolumes of cocompact lattices in SL_3(K). Our proofs combine a construction of Cartwright and Steger with results about Singer cycles and their normalisers, and geometric arguments.

Cocompact lattices on \tilde{A}_n buildings

Using Singer polygons, we construct locally finite affine buildings of types ~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the ~C_2-buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type ~C_2. Integral and rational group homology for the lattices is also calculated.

/pdf/a-geometric-construction-of-panel-regular-lattices-in-1cbiy8isiw.pdf

A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

This is the author's PhD thesis, published at the Universit\"at M\"unster, Germany in 2010. It contains a detailed description of the results of arXiv:0903.1989, arXiv:0905.0071 and arXiv:0908.2713.

Buildings, Group Homology and Lattices

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A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

Buildings, Group Homology and Lattices

Buildings, Group Homology and Lattices