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Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.
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The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays accumulate onto geodesic laminations which are in some sense filling, but without strong enough properties to correspond to points in the boundary of the loop graph. We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.
It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that when $R$ is a subgroup generated by independent random walks in $G$, then $langle H, Rranglecong Hast R$ with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in $G$. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In particular, when $G$ is the mapping class group of a surface and $H$ is a convex cocompact subgroup we show that $langle H, Rrangle$ is convex cocompact and isomorphic to $ Hast R$.
This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.
We study certain polynomial trace identities in the group $SL(2,IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in S 4 and then in S 5 we give a new proof for Gehring and Martins polynomial trace identities for good words, and extend that result to a larger class which is also closed under a semigroup operation inducing polynomial composition. This new approach is through the use of quaternion algebras over indefinites and an associated group of units. We obtain structure theorems for these quaternion algebras which appear to be of independent interest in S 8. Using these quaternion algebras and their units, we consider their relation to discrete subgroups of $SL(2,IC)$ giving necessary and sufficient criteria for discreteness, and another for arithmeticity S 9. We then show that for the groups $IZ_p*IZ_2$, the complement of the closure of roots of the good word polynomials is precisely the moduli space of geometrically finite discrete and faithful representations a result we show holds in greater generality in S12.
We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebes conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramms transboundary extremal lengths.
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