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We give a presentation for the baseleaf preserving mapping class group Mod() of the punctured solenoid . The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that Mod() has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which Mod() acts.doi:10.2140/agt.2007.7.1171 fatcat:zta3prlfp5fdnf6b2uc7jkbhgy