Local decomposability characterises when a piecewise full group admits a unique non-discrete t.d.l.c. topology, and the resulting alternating full group is an open, compactly generated, simple derived subgroup that is universal within the class SLD.
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Locally compact piecewise full groups of homeomorphisms
Local decomposability characterises when a piecewise full group admits a unique non-discrete t.d.l.c. topology, and the resulting alternating full group is an open, compactly generated, simple derived subgroup that is universal within the class SLD.