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We give an explicit construction of a family of lattices in PU (1, 2) originally constructed by Livné. Following Thurston, we construct these lattices as the modular group of certain Euclidean cone metrics on the sphere. We give connections between these groups and other groups of complex hyperbolic isometries.

In our previous paper “The primes contain arbitrarily long polynomial progressions” [$Acta Math$., 201 (2008), 213–305] there were some minor errors in our definition of the polynomial forms and polynomial correlation conditions, which is unreasonably strong as written due to the overly loose bound on the degree of the polynomials involved. In this erratum we repair this issue by capping the degree of the polynomials more strongly, and adjusting some other relevant components of the argument accordingly.

No abstract uploaded!

We investigate whether a Stein manifold$M$which allows proper holomorphic embedding into ℂ^{$n$}can be embedded in such a way that the image contains a given discrete set of points and in addition follow arbitrarily close to prescribed tangent directions in a neighbourhood of the discrete set. The infinitesimal version was proven by Forstnerič to be always possible. We give a general positive answer if the dimension of$M$is smaller than$n$/2 and construct counterexamples for all other dimensional relations. The obstruction we use in these examples is a certain hyperbolicity condition.