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Work on a compact oriented surface, possibly with finitely many marked points, and fix its boundary pointwise during isotopies. Suppose an Alexander system cuts the surface into discs, each containing at most one marked point. This is the filling hypothesis needed here.

A homeomorphism preserving every member up to isotopy can be adjusted by a simultaneous isotopy to preserve its structure graph of an Alexander system. If the induced graph action fixes every vertex and every oriented edge, the remaining maps on the complementary discs are isotopic to the identity by the Alexander trick. Thus the homeomorphism is isotopic to the identity. Since there are only finitely many automorphisms of the finite oriented graph, the pointwise curve-and-arc-class stabilizer is finite.^[ref]

The strong filling condition matters for a surface with boundary. A boundary-parallel Dehn twist can fix every interior curve class while remaining nontrivial relative to the boundary. A system cutting all the way into discs detects this missing information.

Benson Farb works on topology and geometric group theory. His University of Chicago page identifies his research publications and books.^[ref]

With Dan Margalit, he wrote A primer on mapping class groups (Version 5.0) by Benson Farb and Dan Margalit. The book studies symmetries of surfaces up to deformation. A Dehn twist is supported in an annulus, yet repeated twisting has a precisely calculable effect on intersection numbers of curves. This links a local geometric operation with the algebra of a mapping class group.^[ref]

A Dehn twist about a simple closed curve $\gamma$ in an oriented surface is a homeomorphism supported in an annulus around $\gamma$. In the coordinates and orientation convention of a right-handed Dehn twist, it has the formwhere $f$ is zero near one boundary circle and one near the other. It extends by the identity outside the annulus; subtracting $2\pi f(r)$ gives its inverse. Changing the interpolation without changing the end values gives the same mapping class.

For essential simple closed curves $\alpha ,\gamma$ and every integer $n$,Thus $i(\alpha ,\gamma )>0$ forces all iterates to have distinct isotopy classes. In minimal position of curves or arcs, each crossing strand winds $n$ times through the other strands; the bigon criterion proves the counted intersections cannot be removed.^[ref]

On a torus, take curves representing a basis $a,b$ of first homology, with $a\cdot b=1$. Under the stated shear convention, $T_a^n(b)=b+na$, and its geometric intersection number with $b$ is $|n|$. A twist about a curve bounding an unpunctured disc is instead isotopic to the identity by the Alexander trick. Boundary-parallel twists can remain nontrivial when isotopies must fix the boundary pointwise.

Hyperbolic geometry is the geometry of constant negative curvature. Its standard model is hyperbolic space, the complete simply connected Riemannian manifold with sectional curvature $-1$. In two dimensions, the Poincaré half-plane model isIts geodesics are vertical lines and semicircles orthogonal to the real axis. The real axis is an ideal boundary, at infinite hyperbolic distance from every interior point.^[ref]

The parallel postulate fails: through a point off a hyperbolic line, there are infinitely many hyperbolic lines disjoint from it. A hyperbolic triangle has angle sum less than $\pi$, and its hyperbolic area is exactly the angle defect in curvature $-1$. A hyperbolic disc of radius $R$ has area $2\pi (\cosh R-1)$, growing exponentially in $R$. These give concrete ways that hyperbolic geometry differs from the locally Euclidean behavior seen at very small distances.

The normalization fixes a length scale: multiplying the Riemannian metric by $R_0^2$ changes its curvature to $-1/R_0^2$ and multiplies all lengths by $R_0$. Quotients by suitable discrete subgroups of isometries produce hyperbolic surfaces and hyperbolic three-manifolds, connecting the geometry to fundamental groups and topology.

For a connected oriented topological surface of finite type with negative Euler characteristic, every orientation-preserving mapping class has a periodic representative, preserves an essential multicurve, or has a pseudo-Anosov map representative. In the usual disjoint classification, the reducible case means a nonperiodic class preserving such a multicurve. The assertion concerns representatives up to isotopy, rather than requiring an arbitrary homeomorphism in a periodic class to have finite order.

For a torus with one puncture, the action on first homology identifies its mapping class group with $S{L}_2(\mathbb{Z})$. For the corresponding matrix $M$, the periodic classes have $|\mathrm{tr}M|<2$ or $M=\pm I$. Nonperiodic reducible classes have $|\mathrm{tr}M|=2$ and $M\ne \pm I$; pseudo-Anosov map classes have $|\mathrm{tr}M|>2$. For example, $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ is a Dehn twist, while $I$ is periodic despite having the same trace.^[ref]

A puncture is distinguished from a boundary circle fixed pointwise: boundary twists give additional mapping classes invisible to the displayed $S{L}_2(\mathbb{Z})$ action. For the closed torus, the hyperbolic case has an Anosov homeomorphism representative.