Let $\mathbb{C}[G]$ be the group algebra of the finite group $G$ over $\mathbb{C}$. We could consider what happens when considering paths in this vector space, but this doesn't seem too interesting. Instead, we could consider `multiplicative' paths instead.
As an example, take two (for the moment) invertible elements $\alpha_1$ and $\alpha_2$. Since they are both invertible, there exists some $\beta$ such that $\alpha_2 = \beta \alpha_1$. (Under some assumptions) we could split up the $\beta$ into a product of multiple elements $\beta_i \in \mathbb{C}[G]$, so that $\alpha_2 = \left(\Pi_i\beta_i\right) \alpha_1$. By taking the limit of a continuous product and taking all $\beta_i$ to be infinitesimally close to the identity element $1e$, we should now be able to parametrize `multiplicative' paths in our space.
My first question is, is the above well-defined? In particular if none of the $\beta_i$ become non-invertible. Secondly, have these type of paths been analyzed before? They seem to have some nice properties. For example, consider the function $\phi(\alpha) = \mathrm{Det}(\rho(\alpha))$, where $\rho$ is the regular representation of $G$. Then we have that for the path given by $\Pi_{t}\beta(t)$ that
$$\phi(\Pi_{t}\beta(t))\alpha_1 = \phi(\Pi_{t}\beta(t))\alpha_1\\ \rightarrow \phi(\Pi_{t}\beta(t)) = \phi(\beta) = \frac{\phi(\alpha_2)}{\phi(\alpha_1)}.$$
That is, the value of $\phi(\beta(t))$ is independent of the path itself, only on its beginning and endpoints (one should be careful here with crossing over elements that are non-invertible, but I guess one could make sense out of these cases using a form of the Cauchy principal value). One could extend this idea by integrating over the logarithm of $\phi$. This should give back a variant on the Cauchy integral theorem/residue theorem, but I am still working this out.
Edit: As mentioned in the commments, any path over invertible elements can be rewritten as a multiplicative path. I should have formulated my statement differently. Instead of analysing multiplicative paths, I am more interested in the multiplicative parametrisation of a path (if it exists).