The minimum generator count for Γ^4_n and cΓ^4_n is C(n,3)-1, the abelianization of Γ^4_n is (Z/2Z)^{C(n,3)-1}, and Γ^4_5 is infinite and lacks Property (T).
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Minimal generating sets of groups of Kim-Manturov
The minimum generator count for Γ^4_n and cΓ^4_n is C(n,3)-1, the abelianization of Γ^4_n is (Z/2Z)^{C(n,3)-1}, and Γ^4_5 is infinite and lacks Property (T).