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The free group on two generators is exceptional: the group of outer automorphisms of $F(a,b)$ is $GL(2, \mathbb Z)$. That's because every automorphism preserves the commutator $[a,b]$ up to conjugacy. Every outer automorphism of the fundamental group of a surface with boundary that permutes the boundary components up to conjugacy is represented by a homeomorphism of the surface, by a theorem I believe of Nielsen, so outer automorphisms of $F(a,b) \leftrightarrow $ isotopy classes of homeomorphisms of a torus minus an open disk $\leftrightarrow GL(2,\mathbb Z)$ acting on $\mathbb R^2/\mathbb Z^2$ (you can remove the lattice points to make it a punctured torus).

A free homotopy class of curves $\alpha$ on a surface fills the surface is for any other non-peripheral homotopy class $\beta$, it is impossible to homotope $\alpha$ and $\beta$ so they don't intersect. Surfaces of negative Euler characteristic such as the punctured torus have complete Riemannian metrics of constant negative curvature (hyperbolic structures), where every free homotopy class is represented by a unique geodesic. In a hyperbolic structure $\alpha$ fills the surface if and only if every component of the surface minus $\alpha$ is either simply-connected, or it retracts to an arbitrarily small neighborhood of a puncture.

The rubber band theorem of Steve Kerckhoff says that for a homotopy class $\alpha$ that fills a surface $S$ of finite type and negative Euler characteristic, there is a unique hyperbolic structure for which the length of $\alpha$ is minimized: a rubber band following $\alpha$ pulls the surface into a specific shape. (The length of $\alpha$ is a convex function, in a certain sense, in Teichmüller space: this uses either my earthquake theorem, or Scott Wolpert's geometric analysis of the Weil-Peterson metric. This was the key step in Kerckhoff's proof of the Nielsen fixed point theorem which had an infamous history).

In particular, any outer automorphism of the free group that fixes such a conjugacy class is homotopic to an isometry of a particular hyperbolic structure, which implies the group of all such elements is finite. For an $\alpha$ that doesn't fill the surface, a Dehn twist about any non-intersecting simple closed curve generates an infinite group of automorphisms, as in your example. In the figure, the arcs labeled a* and b* keep track of a homorphism to the free group on two generators.

alt text http://dl.dropbox.com/u/5390048/UnfilledTorus.pdf

It's easy to invent lots of homotopy classes like this on the punctured torus, just as you wrap a ribbon around a present. The curve illustrated below cuts the punctured torus into an open annulus around the puncture, and an open disk, so it qualifies. I leave the word description of this curve as an exercise.

alt text http://dl.dropbox.com/u/5390048/gift-torus.pdf

For free groups on more generators, the theory is trickier, but there is a beautiful theory developed by Bestvina and Handel of train track maps representing automorphisms of free groups, that gives a method to determine what elements are fixed up to conjugacy by a particular automorphism. I believe this will give examples in the general case, but I think it takes some work and I haven't thought through the details. A related subject: there's an elaborate theory of when word-hyperbolic groups can have infinite automorphism groups, via the theory of group actions on $\mathbb R$-trees; and your question is close to a special case of this, for one-relator groups.

The free group on two generators is exceptional: the group of outer automorphisms of $F(a,b)$ is $GL(2, \mathbb Z)$. That's because every automorphism preserves the commutator $[a,b]$ up to conjugacy. Every outer automorphism of the fundamental group of a surface with boundary that permutes the boundary components up to conjugacy is represented by a homeomorphism of the surface, by a theorem I believe of Nielsen, so outer automorphisms of $F(a,b) \leftrightarrow $ isotopy classes of homeomorphisms of a torus minus an open disk $\leftrightarrow GL(2,\mathbb Z)$ acting on $\mathbb R^2/\mathbb Z^2$ (you can remove the lattice points to make it a punctured torus).

A free homotopy class of curves $\alpha$ on a surface fills the surface is for any other non-peripheral homotopy class $\beta$, it is impossible to homotope $\alpha$ and $\beta$ so they don't intersect. Surfaces of negative Euler characteristic such as the punctured torus have complete Riemannian metrics of constant negative curvature (hyperbolic structures), where every free homotopy class is represented by a unique geodesic. In a hyperbolic structure $\alpha$ fills the surface if and only if every component of the surface minus $\alpha$ is either simply-connected, or it retracts to an arbitrarily small neighborhood of a puncture.

The rubber band theorem of Steve Kerckhoff says that for a homotopy class $\alpha$ that fills a surface $S$ of finite type and negative Euler characteristic, there is a unique hyperbolic structure for which the length of $\alpha$ is minimized: a rubber band following $\alpha$ pulls the surface into a specific shape. (The length of $\alpha$ is a convex function, in a certain sense, in Teichmüller space: this uses either my earthquake theorem, or Scott Wolpert's geometric analysis of the Weil-Peterson metric. This was the key step in Kerckhoff's proof of the Nielsen fixed point theorem which had an infamous history).

In particular, any outer automorphism of the free group that fixes such a conjugacy class is homotopic to an isometry of a particular hyperbolic structure, which implies the group of all such elements is finite. For an $\alpha$ that doesn't fill the surface, a Dehn twist about any non-intersecting simple closed curve generates an infinite group of automorphisms, as in your example. In the figure, the arcs labeled a* and b* keep track of a homorphism to the free group on two generators.

alt text http://dl.dropbox.com/u/5390048/UnfilledTorus.png

It's easy to invent lots of homotopy classes like this on the punctured torus, just as you wrap a ribbon around a present. The curve illustrated below cuts the punctured torus into an open annulus around the puncture, and an open disk, so it qualifies. I leave the word description of this curve as an exercise.

alt text http://dl.dropbox.com/u/5390048/gift-torus.png

For free groups on more generators, the theory is trickier, but there is a beautiful theory developed by Bestvina and Handel of train track maps representing automorphisms of free groups, that gives a method to determine what elements are fixed up to conjugacy by a particular automorphism. I believe this will give examples in the general case, but I think it takes some work and I haven't thought through the details. A related subject: there's an elaborate theory of when word-hyperbolic groups can have infinite automorphism groups, via the theory of group actions on $\mathbb R$-trees; and your question is close to a special case of this, for one-relator groups.

The free group on two generators is exceptional: the group of outer automorphisms of $F(a,b)$ is $GL(2, \mathbb Z)$. That's because every automorphism preserves the commutator $[a,b]$ up to conjugacy. Every outer automorphism of the fundamental group of a surface with boundary that permutes the boundary components up to conjugacy is represented by a homeomorphism of the surface, by a theorem I believe of Nielsen, so outer automorphisms of $F(a,b) \leftrightarrow $ isotopy classes of homeomorphisms of a torus minus an open disk $\leftrightarrow GL(2,\mathbb Z)$ acting on $\mathbb R^2/\mathbb Z^2$ (you can remove the lattice points to make it a punctured torus).

A free homotopy class of curves $\alpha$ on a surface fills the surface is for any other non-peripheral homotopy class $\beta$, it is impossible to homotope $\alpha$ and $\beta$ so they don't intersect. Surfaces of negative Euler characteristic such as the punctured torus have complete Riemannian metrics of constant negative curvature (hyperbolic structures), where every free homotopy class is represented by a unique geodesic. In a hyperbolic structure $\alpha$ fills the surface if and only if every component of the surface minus $\alpha$ is either simply-connected, or it retracts to an arbitrarily small neighborhood of a puncture.

The rubber band theorem of Steve Kerckhoff says that for a homotopy class $\alpha$ that fills a surface $S$ of finite type and negative Euler characteristic, there is a unique hyperbolic structure for which the length of $\alpha$ is minimized: a rubber band following $\alpha$ pulls the surface into a specific shape. (The length of $\alpha$ is a convex function, in a certain sense, in Teichmüller space: this uses either my earthquake theorem, or Scott Wolpert's geometric analysis of the Weil-Peterson metric. This was the key step in Kerckhoff's proof of the Nielsen fixed point theorem which had an infamous history).

In particular, any outer automorphism of the free group that fixes such a conjugacy class is homotopic to an isometry of a particular hyperbolic structure, which implies the group of all such elements is finite.

It's easy to invent lots of homotopy classes like this on the punctured torus, just as you wrap a ribbon around a present. The curve illustrated below cuts the punctured torus into an open annulus around the puncture, and an open disk, so it qualifies. I leave the word description of this curve as an exercise.

alt text http://dl.dropbox.com/u/5390048/gift-torus.pdf

For free groups on more generators, the theory is trickier, but there is a beautiful theory developed by Bestvina and Handel of train track maps representing automorphisms of free groups, that gives a method to determine what elements are fixed up to conjugacy by a particular automorphism. I believe this will give examples in the general case, but I think it takes some work and I haven't thought through the details. A related subject: there's an elaborate theory of when word-hyperbolic groups can have infinite automorphism groups, via the theory of group actions on $\mathbb R$-trees; and your question is close to a special case of this, for one-relator groups.

The free group on two generators is exceptional: the group of outer automorphisms of $F(a,b)$ is $GL(2, \mathbb Z)$. That's because every automorphism preserves the commutator $[a,b]$ up to conjugacy. Every outer automorphism of the fundamental group of a surface with boundary that permutes the boundary components up to conjugacy is represented by a homeomorphism of the surface, by a theorem I believe of Nielsen, so outer automorphisms of $F(a,b) \leftrightarrow $ isotopy classes of homeomorphisms of a torus minus an open disk $\leftrightarrow GL(2,\mathbb Z)$ acting on $\mathbb R^2/\mathbb Z^2$ (you can remove the lattice points to make it a punctured torus).

A free homotopy class of curves $\alpha$ on a surface fills the surface is for any other non-peripheral homotopy class $\beta$, it is impossible to homotope $\alpha$ and $\beta$ so they don't intersect. Surfaces of negative Euler characteristic such as the punctured torus have complete Riemannian metrics of constant negative curvature (hyperbolic structures), where every free homotopy class is represented by a unique geodesic. In a hyperbolic structure $\alpha$ fills the surface if and only if every component of the surface minus $\alpha$ is either simply-connected, or it retracts to an arbitrarily small neighborhood of a puncture.

The rubber band theorem of Steve Kerckhoff says that for a homotopy class $\alpha$ that fills a surface $S$ of finite type and negative Euler characteristic, there is a unique hyperbolic structure for which the length of $\alpha$ is minimized: a rubber band following $\alpha$ pulls the surface into a specific shape. (The length of $\alpha$ is a convex function, in a certain sense, in Teichmüller space: this uses either my earthquake theorem, or Scott Wolpert's geometric analysis of the Weil-Peterson metric. This was the key step in Kerckhoff's proof of the Nielsen fixed point theorem which had an infamous history).

In particular, any outer automorphism of the free group that fixes such a conjugacy class is homotopic to an isometry of a particular hyperbolic structure, which implies the group of all such elements is finite. For an $\alpha$ that doesn't fill the surface, a Dehn twist about any non-intersecting simple closed curve generates an infinite group of automorphisms, as in your example. In the figure, the arcs labeled a* and b* keep track of a homorphism to the free group on two generators.

alt text http://dl.dropbox.com/u/5390048/UnfilledTorus.pdf

It's easy to invent lots of homotopy classes like this on the punctured torus, just as you wrap a ribbon around a present. The curve illustrated below cuts the punctured torus into an open annulus around the puncture, and an open disk, so it qualifies. I leave the word description of this curve as an exercise.

alt text http://dl.dropbox.com/u/5390048/gift-torus.pdf

For free groups on more generators, the theory is trickier, but there is a beautiful theory developed by Bestvina and Handel of train track maps representing automorphisms of free groups, that gives a method to determine what elements are fixed up to conjugacy by a particular automorphism. I believe this will give examples in the general case, but I think it takes some work and I haven't thought through the details. A related subject: there's an elaborate theory of when word-hyperbolic groups can have infinite automorphism groups, via the theory of group actions on $\mathbb R$-trees; and your question is close to a special case of this, for one-relator groups.

The free group on two generators is exceptional: the group of outer automorphisms of $F(a,b)$ is $SL(2, \mathbb Z)$. That's because every automorphism preserves the commutator $[a,b]$ up to conjugacy. Every outer automorphism of the fundamental group of a surface with boundary that permutes the boundary components up to conjugacy is represented by a homeomorphism of the surface, by a theorem I believe of Nielsen, so outer automorphisms of $F(a,b) \leftrightarrow $ isotopy classes of homeomorphisms of a torus minus an open disk $\leftrightarrow SL(2,\mathbb Z)$ acting on $\mathbb R^2/\mathbb Z^2$ (you can remove the lattice points to make it a punctured torus).

A free homotopy class of curves $\alpha$ on a surface fills the surface is for any other non-peripheral homotopy class $\beta$, it is impossible to homotope $alpha$ and $\beta$ so they don't intersect. Surfaces of negative Euler characteristic such as the punctured torus have complete Riemannian metrics of constant negative curvature (hyperbolic structures), where every free homotopy class is represented by a unique geodesic. In a hyperbolic structure $\alpha$ fills the surface if and only if every component of the surface minus $\alpha$ is either simply-connected, or it retracts to an arbitrarily small neighborhood of a puncture.

The rubber band theorem of Steve Kerckhoff says that for a homotopy class $\alpha$ that fills a surface $S$ of finite type and negative Euler characteristic, there is a unique hyperbolic structure for which the length of $\alpha$ is minimized: a rubber band following $\alpha$ pulls the surface into a specific shape. (The length of $\alpha$ is a convex function, in a certain sense, in Teichmüller space: this uses either my earthquake theorem, or Scott Wolpert's geometric analysis of the Weil-Peterson metric. This was the key step in Kerckhoff's proof of the Nielsen fixed point theorem which had an infamous history).

In particular, any outer automorphism of the free group that fixes such a conjugacy class is homotopic to an isometry of a particular hyperbolic structure, which implies the group of all such elements is finite.

It's easy to invent lots of homotopy classes like this on the punctured torus, just as you wrap a ribbon around a present. The curve illustrated below cuts the punctured torus into an open annulus around the puncture, and an open disk, so it qualifies. I leave the word description of this curve as an exercise.

alt text http://dl.dropbox.com/u/5390048/gift-torus.pdf

For free groups on more generators, the theory is trickier, but there is a beautiful theory developed by Bestvina and Handel of train track maps representing automorphisms of free groups, that gives a method to determine what elements are fixed up to conjugacy by a particular automorphism. I believe this will give examples in the general case, but I think it takes some work and I haven't thought through the details. A related subject: there's an elaborate theory of when word-hyperbolic groups can have infinite automorphism groups, via the theory of group actions on $\mathbb R$-trees; and your question is close to a special case of this, for one-relator groups.

The free group on two generators is exceptional: the group of outer automorphisms of $F(a,b)$ is $GL(2, \mathbb Z)$. That's because every automorphism preserves the commutator $[a,b]$ up to conjugacy. Every outer automorphism of the fundamental group of a surface with boundary that permutes the boundary components up to conjugacy is represented by a homeomorphism of the surface, by a theorem I believe of Nielsen, so outer automorphisms of $F(a,b) \leftrightarrow $ isotopy classes of homeomorphisms of a torus minus an open disk $\leftrightarrow GL(2,\mathbb Z)$ acting on $\mathbb R^2/\mathbb Z^2$ (you can remove the lattice points to make it a punctured torus).

A free homotopy class of curves $\alpha$ on a surface fills the surface is for any other non-peripheral homotopy class $\beta$, it is impossible to homotope $\alpha$ and $\beta$ so they don't intersect. Surfaces of negative Euler characteristic such as the punctured torus have complete Riemannian metrics of constant negative curvature (hyperbolic structures), where every free homotopy class is represented by a unique geodesic. In a hyperbolic structure $\alpha$ fills the surface if and only if every component of the surface minus $\alpha$ is either simply-connected, or it retracts to an arbitrarily small neighborhood of a puncture.

The rubber band theorem of Steve Kerckhoff says that for a homotopy class $\alpha$ that fills a surface $S$ of finite type and negative Euler characteristic, there is a unique hyperbolic structure for which the length of $\alpha$ is minimized: a rubber band following $\alpha$ pulls the surface into a specific shape. (The length of $\alpha$ is a convex function, in a certain sense, in Teichmüller space: this uses either my earthquake theorem, or Scott Wolpert's geometric analysis of the Weil-Peterson metric. This was the key step in Kerckhoff's proof of the Nielsen fixed point theorem which had an infamous history).

In particular, any outer automorphism of the free group that fixes such a conjugacy class is homotopic to an isometry of a particular hyperbolic structure, which implies the group of all such elements is finite.

It's easy to invent lots of homotopy classes like this on the punctured torus, just as you wrap a ribbon around a present. The curve illustrated below cuts the punctured torus into an open annulus around the puncture, and an open disk, so it qualifies. I leave the word description of this curve as an exercise.

alt text http://dl.dropbox.com/u/5390048/gift-torus.pdf

For free groups on more generators, the theory is trickier, but there is a beautiful theory developed by Bestvina and Handel of train track maps representing automorphisms of free groups, that gives a method to determine what elements are fixed up to conjugacy by a particular automorphism. I believe this will give examples in the general case, but I think it takes some work and I haven't thought through the details. A related subject: there's an elaborate theory of when word-hyperbolic groups can have infinite automorphism groups, via the theory of group actions on $\mathbb R$-trees; and your question is close to a special case of this, for one-relator groups.