# Research

## a summary of my research topics

### 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

1. the word problem as $\mathrm{W}(G,X) = \left\{ w \in X^\ast \, : \, w = 1_G \right\};$
2. the co-word problem as $\mathrm{coW}(G,X) = \left\{ w \in X^\ast \, : \, w \neq 1_G \right\};$
3. 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?