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Abstract We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups. 

Abstract We construct an unfolding path in Outer space which does not converge in the boundary, and instead it accumulates on the entire 1-simplex of projectivized length measures on a nongeometric arational$${\mathbb R}$$-treeT. We also show thatTadmits exactly two dual ergodic projective currents. This is the first nongeometric example of an arational tree that is neither uniquely ergodic nor uniquely ergometric. 

We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends spaces, as defined by Mann and Rafi, and with 0 or infinite genus, we show that when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property. 

ABSTRACT Permafrost microbial research has flourished in the past decades, due in part to improvements in sampling and molecular techniques, but also the increased focus on the permafrost greenhouse gas feedback to climate change and other ecological processes in high latitude and alpine permafrost soils. Permafrost microorganisms are adapted to these extreme environments and remain active at low temperatures and when resources are limited. They are also an important component of global elemental cycles as they regulate organic matter turnover and greenhouse gas production, particularly as permafrost thaws. Here we review the permafrost microbiology literature coupled with an exploration of its historical aspects, with a particular focus on a new understanding advanced by molecular biology techniques. We further identify knowledge gaps and ways forward to improve our understanding of microbial contributions to ecosystem biogeochemistry of permafrost‐affected systems. 

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $$S$$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $$S$$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.