Return to Answer

The following was inspired by Jesse Peterson's answer.

Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space. Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.

Any non-commuting pair of elements whose axes share an end point now provide an example of the kind required. It isn't too difficult to construct explicit examples.

UPDATE

As Torsten Ekedhal points out, the last sentence should have read 'It's impossible to construct such examples'! In particular, this doesn't work.

The following was inspired by Jesse Peterson's answer.

Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space. Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.

Any non-commuting pair of elements whose axes share an end point now provide an example of the kind required. It isn't too difficult to construct explicit examples.

UPDATE

As Torsten Ekedahl points out, the last sentence should have read 'It's impossible to construct such examples'! In particular, this doesn't work.

The following was inspired by Jesse Peterson's answer.

Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space. Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.

Any non-commuting pair of elements whose axes share an end point now provide an example of the kind required. It isn't too difficult to construct explicit examples.

The following was inspired by Jesse Peterson's answer.

Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space. Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.

Any non-commuting pair of elements whose axes share an end point now provide an example of the kind required. It isn't too difficult to construct explicit examples.

UPDATE

As Torsten Ekedhal points out, the last sentence should have read 'It's impossible to construct such examples'! In particular, this doesn't work.

The following was inspired by Jesse Peterson's answer.

Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space. Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.

Any pair of elements whose axes share an end point now provide an example of the kind required.

The following was inspired by Jesse Peterson's answer.

Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space. Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.

Any non-commuting pair of elements whose axes share an end point now provide an example of the kind required. It isn't too difficult to construct explicit examples.