Lattices, Trees and Group Actions

给出了保证树格存在的充要条件，即局部有限树的自同构群中有限余体积的离散子群，完整地回答了Bass-Lubotzky关于树格存在的猜想.其目的是研究非阿基米德域上的1阶李群的树格和格之间的联系。研究人员对秩为1的李群中包含的非一致格的显式构造感兴趣，并将它们与一般树格、余体积、算术性和可公度性问题以及Hausdorff维度进行比较。关于李群中的格，A.Lubotzky证明了非阿基米德局部域上的秩为1的李群包含不可数的许多格的共轭类。研究人员得到了Lubotzky格的形变空间的拓扑描述。现在的目的是研究这些变形空间的解析和代数几何结构，并构造非均匀格子的无限维形变空间。这位研究者和H.Garland已经证明了有限域上的Kac-Moody群包含格。现在的目的是证明在秩2中存在格的变形空间，并研究变形空间的结构，非均匀格的基本域的结构，以及在秩2中一致格的可公度群的结构。我们还考虑了在较高阶中的一致格和球形建筑，同余子群，以及在非分裂和广义Kac-Moody群中的格的存在。这位研究者和她的同事们的目标是，在E.Rips的工作之后，通过等距、等距和同伦给出R-树上具有自由或稳定作用的群的分类定理。我们的策略是对R的等距伪群进行分类，然后将其与最近发展起来的用于重构群作用的结构理论相结合。我们正在研究无限树，即没有闭路的连通图，以及它们的对称性的代数结构。具有‘离散’和‘有限体积’的代数结构特别重要。我们已经证实了这种对称性的存在，并正在研究它们的性质。这使我们能够研究数学与物理之间的相互作用。我们的技术在代数、几何和拓扑学中也有应用。

The investigator has determined the necessary and sufficient conditions that ensure the existence of tree lattices, that is discrete subgroup of finite covolume in the automorphism group of a locally finite tree, giving a complete answer to the Bass-Lubotzky conjectures for the existence of tree lattices. The intention is to explore the connections between tree lattices and lattices in rank 1 Lie groups over non-archimedean fields. The investigator is interested in making explicit constructions of non-uniform lattices contained within rank 1 Lie groups, comparing them with general tree lattices, covolumes, questions of arithmeticity and commensurability, and Hausdorff dimension. Concerning lattices in Lie groups, A. Lubotzky showed that rank 1 Lie groups over non-archimedean local fields contain uncountably many conjugacy classes of lattices. The investigator has obtainted a topological description of Lubotzky's deformation spaces of lattices. The aim now is to investigate the analytic and algebro-geometric structure of these deformation spaces, and to construct infinite dimensional deformation spaces for non-uniform lattices. The investigator and H. Garland have established that Kac-Moody groups over finite fields contain lattices. The aim is now to show that in rank 2 there are deformation spaces of lattices and to investigate the structure of the deformation spaces, of fundamental domains for non-uniform lattices, and of commensurability groups of uniform lattices in rank 2. Under consideration also is the existence of uniform lattices and spherical buildings in higher rank, congruence subgroups, and lattices in non-split and generalized Kac-Moody groups. The investigator and her colleagues aim, following the work of E. Rips, to give classification theorems for groups with free or stable actions on R-trees by isometries, and by isometries and homothety. The strategy is to give a classification of the pseudogroups of isometries of R, and then to combine this with a structure theory for reconstructing group actions which has recently been developed.We are studying infinite 'trees', which are connected graphs with no closed circuits, and the algebraic structure of their symmetries. The algebraic structures that are 'discrete' and have 'finite volume' are of particular importance. We have established the existence of such symmetries, and we are investigating their properties. This allows us to study the interactions of mathematics with physics. Our techniques also have applications in algebra, geometry and topology.