Constructing pseudo-Anosov maps with given dilatations (Q2634844)

The authors give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface, and, then, give an explicit construction of the surface and the map when the sufficient condition is met. The main result is the following: if \(\lambda\) is the leading eigenvalue of a nonsingular, aperiodic, odd-block, \(\{ 0, 1 \}\)-matrix \(M\), such that \(M\) satisfies the one-sided condition, and \((M, \epsilon)\) satisfies the alignment condition, then: (i) \(\lambda\) is the dilatation of a pseudo-Anosov map, \(\psi\), of a compact surface, \(S_g\), of genus \(g \leq \frac{1}{2} \dim (M)\); (ii) \(\psi\) is orientation-preserving (resp. reversing) when \(\epsilon\) is 1 (resp. \(-1\)); (iii) When \(g = \frac{1}{2} \dim (M)\) a basis for \(H_1(S_g)\) may be chosen so that the action of \(\psi\) on \(H_1(S_g)\) is given by \(M\). Then \(\chi (M)\) is palindromic or anti-palindromic depending on whether \(\epsilon\) is 1 or \(-1\).