arXiv subjects
Publications and source records attributed to Zhaoshen Zhai.
We present an exposition of the Auinger-Steinberg proof of the Ribes-Zalesskiĭ product theorem for pro-V topologies, where V is a pseudovariety of groups closed under extensions with abelian kernel. This proof is self-contained and is accessible to those acquainted with coverings of graphs, and as such, it provides an easy entry point to various other deep theorems for which the product theorem is formally equivalent to.
We construct a system of 33 essential simple closed curves that are pairwise non-homotopic and intersect at most once on the oriented, closed surface of genus 3. Moreover, we show that our construction is saturated, in the sense that it is not properly contained in any other such system of curves.
We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a quasi-tree. More generally, we show that if each component of a locally-finite Borel graph admits a finitely-separating Borel family of cuts, then we may 'canonically' replace each component of the graph by a tree of special ultrafilter-like objects on cuts called orientations; moreover, if the cuts are dense towards ends, then the union of these trees is a Borel treeing.