# zbMATH — the first resource for mathematics

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
Full Text: