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We consider the asymptotic behavior of properly embedded minimal surfaces inthe product of the hyperbolic plane with the line, taking into account the factthat there is more than one natural compactification of this space. Thisprovides a better setting in which to consider the general problem ofdetermining which curves at infinity are the asymptotic boundary of suchminimal surfaces. We also construct some new examples of such surfaces anddescribe the boundary regularity.

On the asymptotic behavior of minimal surfaces in H${}^2$$\times$R