I'll turn YCor's comment (above) into an answer. Surface groups are the simplest (and historically first) non-free example. Let $S_g$ be the surface of genus $g$. Then the fundamental group has presentation

$\pi_1(S_g) \cong \langle a_1, a_2, \ldots, a_{2g-1}, a_{2g} \mid a_1 a_2 \ldots a_{2g} A_1 A_2 \ldots A_{2g} \rangle$.

This presentation is a Dehn presentation. A variant of this fact is due to Dehn. See the following post for discussion and references.

I'll turn YCor's comment (above) into an answer. Surface groups are the simplest (and historically first) non-free example. Let $S_g$ be the surface of genus $g$. Then the fundamental group has presentation

$\pi_1(S_g) \cong \langle a_1, a_2, \ldots, a_{2g-1}, a_{2g} \mid a_1 a_2 \ldots a_{2g} A_1 A_2 \ldots A_{2g} \rangle$.

This presentation is a Dehn presentation. A variant of this fact is due to Dehn. See the following post for discussion and references.

Here is the simplest (and historically first) answer. Let $S_g$ be the surface of genus $g$. Then the fundamental group has presentation

$\pi_1(S_g) \cong \langle a_1, a_2, \ldots, a_{2g-1}, a_{2g} \mid a_1 a_2 \ldots a_{2g} A_1 A_2 \ldots A_{2g} \rangle$.

This presentation is a Dehn presentation. A variant of this fact is due to Dehn. See the following post for a discussion and references.

I'll turn YCor's comment (above) into an answer. Surface groups are the simplest (and historically first) non-free example. Let $S_g$ be the surface of genus $g$. Then the fundamental group has presentation

$\pi_1(S_g) \cong \langle a_1, a_2, \ldots, a_{2g-1}, a_{2g} \mid a_1 a_2 \ldots a_{2g} A_1 A_2 \ldots A_{2g} \rangle$.

This presentation is a Dehn presentation. A variant of this fact is due to Dehn. See the following post for discussion and references.