Faithful actions of the free group on two generators

Question

I'm trying to come up with faithful actions of the free group on two generators, $G$.

By Cayley's theorem, $G$ acts faithfully on itself by left (or right) multiplication. There are also variants of this action, such as $G$ acting on its own Cayley graph.

I suspect (but don't have a proof yet) that we can also take a construction similar to those described in the answer to this MO question and use the following:

1. Let $a$ be a rotation by $1$ radian around the $z$-axis.
2. Let $b$ be a rotation by $1$ radian around the $x$-axis.

I'm curious what other faithful actions of the free group on two generators exist.

Answer 1

Since the group generated by the two matrices $$ a= \begin{pmatrix} 1 & \lambda\\ 0 & 1 \end{pmatrix},\ b= \begin{pmatrix} 1 & 0\\ \lambda & 1 \end{pmatrix},\ \lambda\in\mathbb{R}, \lambda\geq2, $$ is a free group, the corresponding action of the free group on $\mathbb{R}^2$ is faithful.

Answer 2

Here's a faithful representation on $\mathbb{Z}[\pi] \times \mathbb{Z}[\pi]$.

Let $\vartriangleright$ denote our group action.

Let $a \vartriangleright (x, y)$ be $(x + \pi, y)$.

Let $b \vartriangleright (x, y)$ when $x \in \mathbb{Z}$ be $(x, y+\pi)$.

Let $b \vartriangleright (x, y)$ when $x \not\in \mathbb{Z}$ be $(x, y)$.

We observe that $ab \neq ba$.

$ab \vartriangleright (0,0)$ is $(\pi, \pi)$.

$ba \vartriangleright (0,0)$ is $(0, \pi)$.

Consider a word $\vec{w}$ such as $abb^{-1}$; words do not have to be in simplest form. A word is the product of generators and inverses of generators.

I defined a word to be a quasipalindrome if it is equal to itself reversed and with each generator inverted.

For example, $abb^{-1}a^{-1}$ is a quasipalindrome.

$aba^{-1}$ is not a quasipalindrome because it maps to $a^{-1}b^{-1}a$.

A word in a free group is invertible if and only if it is a quasipalindrome.

By inspection, if $w$ has length less than or equal to $4$, then if $w$ is an identity element then $w$ is a quasipalindrome.

For the general case, suppose I have a word $w$ that is not a quasipalindrome. Let $n$ be the number of times that the string $b$ or $b^{-1}$ appears in $w$.

$w$ fails to fix at least one of $(-n\pi, 0), \cdots, (-\pi, 0), (0, 0), (\pi, 0), \cdots, (n\pi, 0)$.