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Fundamental groups of log configuration spaces and the cuspidalization problem

In the present paper, we study the cuspidalization problem of the fundamental group of a curve by means of the log geometry of the log configuration space, which is a natural compactification of the usual configuration space of the curve. The goal of this paper is to show that the fundamental group of the configuration space is generated by the images from morphisms from a group extension of the fundamental groups of the configuration spaces of lower dimension, and that the fundamental group of the configuration space can be partially reconstructed from a collection of data concerning the fundamental groups of the configuration spaces of lower dimension.