For a review of basic group theory, faithful representations, and Ping Pong, please see the background section for our previous work here. Notice that the lemma will only work prove that a representation is faithful if our group is free. In this project, we extend this to any group with an automatic structure.
Automatic Structures
An automatic structure is a finite-state automata whose paths describe exactly the reduced words in a particular group. For example, we can construct an automatic structure for a free group with four verticies and directed edges labeled \(a, b, a^{-1}, b^{-1}\).
Figure 1: An automatic structure for a free group
Automatic structures exist for other groups as well, such as the cyclic free product \(<a, b | a^2 = b^3 = 1>\). In fact, many groups have multiple representations as automatic structures, the following three all working for this cyclic free product:
Figure 2: Three different automatic structures for a cyclic free product with order 2 and 3
Geodesic Automata
Suppose a group \(G\) is generated by the finite set of generators \(a_{1},a_{2},...,a_{n}\). These generators form an alphabet which naturally forms a language whose words are elements in the group \(G\). In the case of a free group this language corresponds to all possible combinations of elements, but in the case of a finitely presented group with some relations the language corresponds to reduced words i.e. if \(a_{2}^{2} = 1\) then the combination \(a_{1}a_{2}^{2}a_{3}\) is not a word in our language because it can be reduced to \(a_{1}a_{3}\).
Definition: A language \(L\) on an alphabet \(A\) is called a regular language if it is exactly the set of words accepted by a finite-state automaton.
Now suppose we have a group \(G = \langle S \> | \> R \rangle\) where \(S\) is a finite generating set and \(R\) is a finite set of relations. We define the length of an element \(\alpha \in G\) as the minimum length of a word in \(S\) which is equivalent to \(\alpha\). We denote the length of \(\alpha\) by \(|\alpha|\). The distance between any two elements \(\alpha,\beta \in G\) is the length of the group element \(\alpha^{-1}\beta\). We denote this distance by \(d(\alpha,\beta)\).
Definition: A geodesic word in \(G\) is a word in \(S\) whose length is the same as the group element it represents.
For the purpose of this project we will be focusing our attention on a subset of automatic groups whose finite state automata only accepts paths which correspond to geodesic words. These are called Strongly Geodesically Automatic Groups.
Definition: A finitely presented group \(G = \langle S \> | \> R \rangle\) is strongly geodesically automatic if both of the following conditions are satisfied:
For some finite generating set \(S\) in \(G\), the set of geodesic words (with respect to \(S\)) is a regular language.
There is a fixed constant \(D > 0\) so that for any two geodesic words
\(s_{1} \cdots s_{k} = r_{1} \cdots r_{k} = g\)
we have
\(d(s_{1} \cdots s_{j}, r_{1}\cdots r_{j}) < D\)
for all \(1 \leq j \leq k\).
Generalized Ping-Pong
For this project, our focus will be on implementing a generalization of the Ping-Pong Lemma which can be applied to representations of strongly geodesically automatic groups which aren't necessarily free. In order to make this construction we will need one more tool however, which is the Hilbert Metric on \(\mathbb{RP}^{1}\).
Definition: Given a projective chart \(P: \mathbb{RP}^{1} \rightarrow \mathbb{R} \cup \lbrace \infty \rbrace\) and an open interval \(I \subset \mathbb{RP}^{1}\), the Hilbert Metric on \(I\), \(d_{I}\) is defined as follows: If \(a,b\) are endpoints of \(I\), then
\(d_{I}(x,y) = |\text{Log}[a,x,y,b]|\) for all distinct \( x,y \in I\)
where \([a,x,y,b]\) is defined as the cross ratio \(\frac{ c-a}{b-a} \cdot \frac{d-b}{d-c}\).
An important property of the Hilbert metric is as follows: If \( A \in \text{SL}(2,\mathbb{R})\) satisfies \(A(I) = J\) then \(A\) takes \(d_{I}\) to \(d_{J}\). If \(J \varsubsetneqq I\) are open intervals then the metric of \(J\) is greater than the metric of \(I\). If, in addition, \(J \Subset I\) then the metric of \(J\) is greater than the metric of \(I\) by a factor of at least \(\lambda(I,J) > 1\).
Generalized Ping Pong: Suppose \(\Gamma\) is a strongly geodesically automatic group with finite generating set \(( a_{1},\cdots,a_{n} )\) and \(\rho: \Gamma \rightarrow SL(2,\mathbb{R})\) is a representation of \(\Gamma\). Now suppose that there exists non-empty subsets \(M_{i} \subset \mathbb{RP}^{1}\) with\( \overline{M_{i}} \neq \mathbb{RP}^{1}\) such that if \(a_{i} \rightarrow a_{j}\) is a path accepted by the finite state automaton attached to \(\Gamma\)
\(\rho(a_{j})(M_{i}) \Subset M_{j}\)
Then \(\text{Ker}(\rho)\) is bounded.
Note: This result is a corollary of Theorem 2.3 from [1] , and as a result the following proof is heavily adapted from the "if" part of the theorem's proof.
Proof: For each generator \(\alpha\), let \(d_{\alpha}\) be the Riemannian metric on \(M_{\alpha}\) which coincides with the Hilbert metric in each of it's components. Let \(K_{\alpha}\) be the closure of the union of the sets \(\rho(\alpha)M_{\beta}\), where\( \beta \rightarrow \alpha\). We can assume that \(K_{\alpha}\) intersects each connected component of \(M_{\alpha}\), because otherwise we can take a smaller \(M_{\alpha}\). Let \(L_{\alpha} \Subset M_{\alpha}\) be an open set containing\( K_{\alpha}\) and with the same number of connected components as \(M_{\alpha}\). Then each component \(M_{\alpha,i}\) of \(M_{\alpha}\) contains a unique component \(L_{\alpha,i}\) of \(L_{\alpha}\). Let \(\lambda = \text{min}_{\alpha,i}\lambda(M_{\alpha,i},L_{\alpha,i})\).
Take an admissible path on our finite state automaton\( a_{I_{0}} \rightarrow a_{I_{1}} \rightarrow \cdots \rightarrow a_{I_{k}}\), and let\( A = \rho(a_{I_{1}}) \cdots \rho(a_{I_{k}})\). If \(u,v\) belong to the same component of \(M_{I_{0}}\) then
\(d_{a_{I_{k}}}(Au,Av) \leq \lambda^{-k}d_{I_{0}}(u,v)\)
The metrics \(d_{\alpha}|L_{\alpha}\) are comparable to the Euclidean metric \(d\) on \(\mathbb{RP}^{1}\). So if \(u,v\) belong to the same component of \(L_{I_{0}}\) we get \(d(Au,Av) \leq C \lambda^{-k}d(u,v)\), where \(C>0\) is some constant. This in turn implies that
\(||A|| \geq C^{-1/2} \lambda^{k/2}\)
If\(A \in \text{Ker}(\rho)\) then \(||A|| = 1\), however since \(\lambda > 1\)for sufficiently large \(k_{0}, ||A|| \geq C^{-1/2} \lambda^{k_{0}/2} > 1\). Therefore words of length \(m \geq k_{0}\) cannot lie in \(\text{Ker}(\rho)\). \(\square\)
Note that the above result not only proves the existence of a bound on \(\text{Ker}(\rho)\) but also gives values \(C, \lambda\) which can be extracted from the particular configuration of subsets \((M_{i})\) to yield an explicit bound.
References
[1] - A. Avila, J. Bochi, J.-C. Yoccoz, Uniformly Hyperbolic Finite-Valued \(SL(2, \mathbb{R})\)-Cocycles, https://arxiv.org/pdf/0808.0133.pdf