Let $\textbf{MCG}(X)$ denote the group of isotopy classes of self-homeomorphisms of a space $X$. When $S$ is an orientable infinite type surface $S$ with no planar ends and without boundary, the extended mapping class group $\textbf{MCG}(S)$ is isomorphic to $\textbf{MCG}\left(\overline{S}\right)$ where $\overline{S}$ is the Freudenthal compactification of $S$. Using this identification, it follows that $\textbf{MCG}(S)$ canonically embeds into $\text{Out}\left(\pi_1\left(\overline{S}\right)\right)$. It remains open if $\textbf{MCG}(S)$ is isomorphic to $\text{Out}\left(\pi_1\left(\overline{S}\right)\right)$ in the spirit of the Dehn-Neilsen-Baer Theorem. A clear difficulty is the fact that $\overline{S}$ is not locally simply connected and $\pi_1\left(\overline{S}\right)$ is uncountable and not free.

In this talk, we will show that $\overline{S}$ is the quotient of $\mathbb{D}^2$ by a countable edge-pairing on $\mathbb{S}^1$. This structural decomposition implies that $\overline{S}$ may constructed by attaching a single 2-cell to a one-dimensional Peano continuum. Using established technology for dealing with fundamental groups of one-dimensional Peano continua, we show that $\pi_1\left(\overline{S}\right)$ is the free product with amalgamation of two locally free groups along an infinite cyclic group. We also show that every automorphism $\phi:\pi_1\left(\overline{S}\right)\to\pi_1\left(\overline{S}\right)$ is induced by a continuous map $f:\overline{S}\to \overline{S}$ that restricts to a homeomorphism on the end set $\overline{S}\backslash S$.