For geometric 3-manifolds the flexible exponent α(M) is 3, 8/3, 2, 1 or 0 according to the model geometry, proved for the Nil case via Legendrian self-maps homotopic to the identity that send fibers to the orthogonal contact plane.
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Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces
For geometric 3-manifolds the flexible exponent α(M) is 3, 8/3, 2, 1 or 0 according to the model geometry, proved for the Nil case via Legendrian self-maps homotopic to the identity that send fibers to the orthogonal contact plane.