Trees and Group Actions

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CARBONE, DMS 98-00604The proposer is studying automorphism groups of locally finite trees, which are locally compact groups, and their discrete subgroups of finite covolume, which are called tree lattices. The proposer has recently discovered how to construct non-uniform tree lattices (and their quotients); these were conjectured by Hyman Bass and Alex Lubotzky to exist. Of particular interest is the case where the tree is homogeneous of two possible degrees,each a power of a prime plus one. For certain degrees of bi-homogeneity, the tree is then the Bruhat-Tits building of a rank 1 simple Lie group over anon-archimedean local field. Broadly speaking, we are interested in constructingnon-uniform lattices contained within such rank 1 Lie groups, comparing them with general tree lattices, further questions of existence of non-uniform lattices, their covolumes and questions of arithmeticity. The proposer is also studying the edge-indexed quotient graphs of non-uniform tree lattices, which give new examples of infinite expanding diagrams.This is a study of infinite 'trees', which are connected graphs in which there are no closed circuits, and the algebraic structure of their symmetries. The proposer has recently discovered that there exist certain symmetries of trees which are locally finite, called 'non-uniform tree lattices', that allow us to collapse the infinite tree into an infinite 'quotient graph' that has finite volume in an appropriate sense. The symmetries of locally finite trees that give finite quotient graphs have been understood for some time, and it is also known that these give rise to infinite families of finite expander graphs, which are fundamental building blocks for certain communication networks in computer science. The construction of non-uniform tree lattices and their quotient graphs, as well as being natural from many mathematical points of view, is desirable in order that there may be new practical applications in the construction of communication networks.