### Group Geodesic Growth

The *geodesic growth function* of a group, \( G \), with respect to a finite symmetric generating set, \( X \), is given as
\[
\Gamma_{G,X}(n)
=
\#
\left\vert
\left\{
w = x_1 x_2 \cdots x_k \in X^\ast
\, : \,
\ell_X(w) = k \leqslant n
\right\}
\right\vert.
\]
Thus, a geodesic growth function counts the number of distinct *geodesic paths* in the group with respect to the word metric.

We say that \( G \) has

*exponential geodesic growth*if \(\Gamma_{G,X}(n)\) is exponential for every \(X\);*polynomal geodesic growth*if \(\Gamma_{G,X}(n)\) is polynomial for some \(X\);- otherwise, we say that \(G\) has
*intermediate geodesic growth*.

The following are open questions.

- Does there exist a group, \(G\), with a finite symmetric generating set \(X\) such that \(\Gamma_{G,X}(n)\) is intermediate?
- Does there exist a group with intermediate geodesic growth?
- Is there a ‘nice’ classification of polynomial geodesic growth?

### Formal Languages in Group Theory

Given a group, \(G\), with a finite symmetric generating set, \(X\), we will define

- the
*word problem*as \[ \mathrm{W}(G,X) = \left\{ w \in X^\ast \, : \, w = 1_G \right\}; \] - the
*co-word problem*as \[ \mathrm{coW}(G,X) = \left\{ w \in X^\ast \, : \, w \neq 1_G \right\}; \] - and the
*full language of geodesics*as \[ \mathrm{Geod}(G,X) = \left\{ w = x_1 x_2 \cdots x_k \in X^\ast \, : \, \ell_X(w) = k \ \ \mathrm{with} \ \ k \in \mathbb{N} \right\}. \]

We will be interested in class of formal languages which are closed under taking inverse word homomorphism, for example, regular language, context-free and some lesser known classes such as ET0L and Indexed.

If we have a class, \(\mathcal{C}\), of formal languages which is closed under inverse word homomorphism, then \(\mathrm{W}(G,X)\) is in the class \(\mathcal{C}\) for some \(X\) if and only if it is in the class \(\mathcal{C}\) for any choice of \(X\). The same can be said of the co-word problem and the full language of geodesics.

The following are open questions.

- Does there exist a group with a word problem that is indexed but not context-free?
- Is every group with an ET0L word problem virtually free?