The singular braids with n strands, n ≥ 3, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by SGn. There has been another generalization of braid groups, denoted by GV Bn, n ≥ 3, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group GV Bn simultaneously generalizes the classical braid group, as well as the virtual braid group on n strands.We investigate the commutator subgroups SG ′ n and GV B ′ n of these generalized braid groups. We prove that SG ′ n is finitely generated if and only if n ≥ 5, and GV B ′ n is finitely generated if and only if n ≥ 4. Further, we show that both SG ′ n and GV B ′ n are perfect if and only if n ≥ 5.

Let T Wn be the twin group on n arcs, n ≥ 2. The group T W m+2 is isomorphic to Grothendieck's m-dimensional cartographical group Cm, m ≥ 1. In this paper we give a finite presentation for the commutator subgroup T W m+2 , and prove that T W m+2 has rank 2m − 1. We derive that T W m+2 is free if and only if m ≤ 3. From this it follows that T W m+2 is word-hyperbolic and does not contain a surface group if and only if m ≤ 3. It also follows that the automorphism group of T W m+2 is finitely presented for m ≤ 3.As applications to the above results, we derive geometric properties of the ambient group T W m+2 . It is clear from the presentation in Theorem 1.1 that for m ≥ 4, T W m+2 contains free abelian subgroups of rank ≥ 2. By [Mou88, Theorem B], this shows that T W m+2 is not word-hyperbolic for m ≥ 4. Whereas from Corollary 1.4 we observe that T W m+2 is virtually free for m ≤ 3; so it is clear that T W m+2 is word-hyperbolic for m ≤ 3. Hence we have the following characterization for word-hyperbolicity of T W m+2 . Corollary 1.5. The group T W m+2 is word-hyperbolic if and only if m ≤ 3.Gordon, Long and Reid proved in [GLR04] that a coxeter group G is virtually free if and only if G does not contain a surface group. Since T W m+2 is finitely generated, by Corollary 1.5, it can not be virtually free for m ≥ 4. Hence we have the following.Corollary 1.6. The group T W m+2 does not contain a surface group if and only if m ≤ 3.

The singular braids with [Formula: see text] strands, [Formula: see text], were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by [Formula: see text]. There has been another generalization of braid groups, denoted by [Formula: see text], [Formula: see text], which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group [Formula: see text] simultaneously generalizes the classical braid group, as well as the virtual braid group on [Formula: see text] strands. We investigate the commutator subgroups [Formula: see text] and [Formula: see text] of these generalized braid groups. We prove that [Formula: see text] is finitely generated if and only if [Formula: see text], and [Formula: see text] is finitely generated if and only if [Formula: see text]. Further, we show that both [Formula: see text] and [Formula: see text] are perfect if and only if [Formula: see text].