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For$$2 \leq d \leq 5$$, we show that the class of the Hurwitz space of smooth degree$$d$$, genus$$g$$covers of$$\mathbb {P}^1$$stabilizes in the Grothendieck ring of stacks as$$g \to \infty$$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.
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For and finite groups, does there exist a 3-manifold group with as a quotient but no as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments.more » « less
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null (Ed.)Abstract We show that, as n goes to infinity, the free group on n generators, modulo {n+u} random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen–Lenstra heuristics. For each n , these random groups belong to the few relator model in the Gromov model of random groups.more » « less
