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Let $$\B_3$$ be the Artin braid group on $$3$$ strands and $$\PB_3$$ be the corresponding pure braid group. In this paper, we construct the groupoid $$\GTSh$$ of $$\GT$$-shadows for a (possibly more tractable) version $$\GTh_0$$ of the Grothendieck-Teichmueller group $$\GTh$$ introduced in paper \cite{HS-fund-groups} by D. Harbater and L. Schneps. We call this group the gentle version of $$\GTh$$ and denote it by $$\GTh_{gen}$$. The objects of $$\GTSh$$ are finite index normal subgroups $$\N$$ of $$\B_3$$ satisfying the condition $$\N \le \PB_3$$. Morphisms of $$\GTSh$$ are called $$\GT$$-shadows and they may be thought of as approximations to elements of $$\GTh_{gen}$$. We show how $$\GT$$-shadows can be obtained from elements of $$\GTh_{gen}$$ and prove that $$\GTh_{gen}$$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $$\GTSh$$. Using this result, we get a criterion for identifying genuine $$\GT$$-shadows. 

We completely classify the orientable infinite-type surfaces S such that PMap(S), the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of surfaces with finitely many ends and no noncompact boundary components, we prove the mapping class group Map(S) does not have automatic continuity. We also completely classify the surfaces such that PMapc (S), the subgroup of the pure mapping class group composed of elements with representatives that can be approximated by compactly supported homeomorphisms, has automatic continuity. In some cases when PMapc (S) has automatic continuity, we show any homomorphism from PMapc (S) to a countable group is trivial. 

We apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic fillings to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings. 

The Alexander method is a combinatorial tool used to determine when two elements of the mapping class group are equal. In this paper we extend the Alexander method to include the case of infinite-type surfaces. Versions of the Alexander method were proven by Hernández--Morales--Valdez, Hernández--Hidber, and Dickmann. As sample applications, we verify a particular relation in the mapping class group, show that the centralizers of many twist subgroups of the mapping class group are trivial, and provide a simple basis for the topology of the mapping class group.