serpentinegroup

Serpentine group is not a standard term in mathematics with a universally accepted definition. When used, it typically refers informally to a finitely presented group whose generators and relations are arranged in a linear, serpentine pattern, evocative of a winding path.

One common interpretive route is to identify a serpentine group with a right-angled Artin group defined by a path graph. In that view, the group G_n is generated by g_1, ..., g_n with commutation relations [g_i, g_j] = 1 whenever i and j are adjacent (that is, |i - j| = 1), and with no relations among non-adjacent generators. This yields a non-abelian, infinite group for n ≥ 3, whose structure reflects a chain-like pattern. The abelianization of such a group is isomorphic to Z^n, and many standard techniques for right-angled Artin groups can be applied to study its subgroups, automorphisms, and geometric properties. The Cayley graph and growth behavior align with the features of RAAGs on path graphs, giving a visually serpentine feel to the defining relations.

Alternatively, the term can be used to suggest other line-like presentations where relations run along a sequence of generators, producing Cayley graphs that resemble a serpentine curve. In scholarly practice, it is more common to specify the exact presentation and explore its consequences, rather than rely on the informal label.

See also: right-angled Artin groups, graph groups, braid groups, geometric group theory.