Groups Acting on Products of Trees
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Groups Acting on Products of Trees


In this thesis, we explore many aspects of groups acting on trees and on products of trees. These ideas are central to the field of geometric group theory, the study of infinite groups by their large-scale behavior. Many of our techniques are algebraic and arithmetic in nature. Most of this work is motivated by the following question:

\begin{ques} If $G$ is the fundamental group of a closed surface, can $G$ act freely on a locally compact product of trees? \end{ques}

In fact, the more general question of whether a hyperbolic group which is not virtually free can act properly on a locally compact Euclidean building is open. Of course such an action gives a cubulation of the group in perhaps the simplest combinatorial type of cube complex, so this is a necessary condition on the group. For instance, which right-angled Coxeter or Artin groups admit proper actions on products of trees?

In a landmark paper of 1987, Gromov defines the class of hyperbolic groups, and nearly every branch of modern geometric group theory is represented in Gromov's original paper. However, Dehn's work in 1910 and 1911 may be viewed as the earliest work in geometric group theory. For several decades, Dehn's work was viewed with a more combinatorial lens, until Gromov emphasized the intimate connection with hyperbolic geometry.

Thurston's work in 3-dimensional topology together with Gromov's hyperbolic groups aligned topology and group theory. One might ask \emph{to what extent is a 3-manifold determined by its fundamental group?} The Poincar e conjecture is a precise version of this question, famously resolved by Perelman. In a sense, the fundamental group of a 3-manifold is an excellent replacement for the manifold itself, and in the finite-volume hyperbolic case, the fundamental group miraculously contains all of the metric information!

For closed hyperbolic surfaces, the fundamental group knows only the genus, and there is a high-dimensional space of representations a given surface can support. For a 4-manifold, the fundamental group says relatively little about the topology or the geometry. Every countable group can occur as the fundamental group of a 4-manifold, and there is a vast land of simply connected four-manifolds. However, it is interesting to note that, at least up to finite index, closed hyperbolic 3-manifold fundamental groups are determined by the data of an automorphism of a surface group. Upon replacing ``hyperbolic 3-manifolds'' with ``discrete subgroups of $\PSL_2(\C)$'', it is tempting to consider ``subgroups of $\PSL_2(\C)$''. This is the approach we take in this thesis. We observe that $\PSL_2(\overline{\Q})$ acts properly discontinuously and cocompactly on a ``restricted'' product of infinitely many hyperbolic planes, 3-spaces, and finite valence trees. By allowing transcendental entries, we get a similar statement, but not all of the trees will have finite-valence. But in fact, subgroups of $\PSL_2(\C)$ with cofinite volume are actually already conjugate into $\PSL_2(\overline{\Q})$ anyway. That is to say, every hyperbolic 3-manifold of finite volume comes equipped with a canonical action on a product of infinitely many finite-valence trees.

One can ask which properties of discrete subgroups of $\PSL_2(\C)$ might carry over to these more general subgroups of $\PSL_2(\C)$. Some celebrated properties of Kleinian groups include coherence, tameness, LERFness. It is unknown to what extent each of these properties might hold in this more general setting.

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