Following the work of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinitetype surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically nontrivial big mapping class group to admit such a quasiisometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe represents the first example of a mapping class group known to be non-elementary hyperbolic.

Abstract. We study random torsion-free nilpotent groups generated by a pair of random words of length ℓ in the standard generating set of Un(Z). Specifically, we give asymptotic results about the step properties of the group when the lengths of the generating words are functions of n. We show that the threshold function for asymptotic abelianness is ℓ = c √ n, for which the probability approaches e −2c 2 , and also that the threshold function for having full-step, the same step as Un(Z), is between cn 2 and cn 3 .

No abstract

We show that the extended based mapping class group of an infinitetype surface is naturally isomorphic to the automorphism group of the loop graph of that surface. Additionally, we show that the extended mapping class group stabilizing a finite set of punctures is isomorphic to the arc graph relative to that finite set of punctures. This extends a known result for sufficiently complex finite-type surfaces, and provides a new angle from which to study the mapping class groups of infinite-type surfaces.

We show that the extended based mapping class group of an infinite-type surface is naturally isomorphic to the automorphism group of the loop graph of that surface. Additionally, we show that the extended mapping class group stabilizing a finite set of punctures is isomorphic to the arc graph relative to that finite set of punctures. This extends a known result for sufficiently complex finite-type surfaces, and provides a new angle from which to study the mapping class groups of infinite-type surfaces.