No abstract

No abstract

No abstract

Using totally symmetric sets, Chudnovsky, Kordek, Li, and Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups, and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.A guiding question to motivate Theorem A is the following conjecture.Conjecture (Margalit [5]). For n ≥ 5, Σ n is the smallest finite, non-cyclic quotient of B n .

In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends, then the grand arc graph is infinite-diameter and δ-hyperbolic. We also show that the mapping class group acts on the grand arc graph by isometries and that the action is quasi-continuous, which is a coarse relaxation of a continuous action. When the surface has stable maximal ends, we also show that this action has finitely many orbits.