Median decompositions arise from any system of vertex cuts via Sageev's dual median graph, are uniquely minimal, satisfy median-width equals clique number on all graphs, and characterize proper geometric actions of groups on median graphs through canonical decompositions of Cayley graphs.
Isometries of
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The double of a virtually compact special Gromov-hyperbolic group along a quasiconvex subgroup is virtually compact special, with a generalization to certain graphs of groups.
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Virtual specialness of the double
The double of a virtually compact special Gromov-hyperbolic group along a quasiconvex subgroup is virtually compact special, with a generalization to certain graphs of groups.