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Algebraic degrees of stretch factors in mapping class groups. (English) Zbl 1356.57017
The author explicitly constructs pseudo-Anosov maps with definite stretch factors on a closed surface of genus \(g\) with orientable foliations.
Let \(S_g\) be a a closed surface of genus \(g\), \(g\geq 2\). Denote by \(c_i\) and \(d_i\), \(1\leq i\leq g\), simple curves on \(S_g\), generating a homological basis of the surface, such that 1) the \(c_i\) pass across the handles of \(S_g\) and are mutually disjoint, 2) the \(d_i\) are also mutually disjoint, 3) each \(c_i\) intersects both the curves \(d_i\) and \(d_{i+1}\) at a single point, \(1\leq i\leq g-1\), and \(c_g\) intersects \(d_g\) at one point, 4) there are no other intersections of the \(c_i\) and \(d_j\). Denote by \(T_\alpha\) the Dehn twist about a curve \(\alpha\).
Let \(T_{A_{g,k}} = (T_{c_{1}}T_{c_{2}}\ldots T_{c_{g-1}} )(T_{c_{g}})^k\), \(T_{{B_g}} = T_{{d_1}}\ldots T_{{d_g}}\), and \(f_{g,k}=T_{A_{g,k}} T_{{B_g}}\). It is proved that \(f_{g,k}\) is a pseudo-Anosov mapping class, the stretch factor \(\lambda (f_{g,k})\) of \(f_{g,k}\) coincides with its homological stretch factor \(\lambda_H (f_{g,k})\); moreover, \(\lambda (f_{g,k})\) is a Salem number, and \(\lim_{g\to \infty}\lambda (f_{g,k})=k-1\).
For the case \(k=4\) it is proved that the minimal polynomial of the stretch factor \(\lambda(f_{g,4})\) is \(x^{2g}-2\sum_{j=1}^{2g-1}x^j+1\), so the algebraic degree of \(\lambda(f_{g,4})\) equals \(2g\). As a corollary, we obtain that for each positive integer \(h\leq g/2\), there is a pseudo-Anosov mapping class such that its stretch factor is a Salem number and has algebraic degree \(2h\).

MSC:
57M50 General geometric structures on low-dimensional manifolds
57M15 Relations of low-dimensional topology with graph theory
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