View source for Binate group

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A group $G$ in which every finitely-generated subgroup $H$ (cf. also [[Finitely-generated group|Finitely-generated group]]) admits a [[Homomorphism|homomorphism]] $\alpha=\alpha_H\colon H\to G$ and an element $u=u_H\in G$ such that for all $h\in H$,

$$h=[\alpha(h),u].$$

(Here, the [[Commutator|commutator]] $[a,b]$ may be taken to mean either $aba^{-1}b^{-1}$ or $a^{-1}b^{-1}ab$.) Thus, the subgroup $H$ is imbedded in $G$ both by $\alpha$ and by inclusion. Such groups $G$ are also called pseudo-mitotic [[#References|[a4]]]. Every binate group is both infinitely generated and an [[Acyclic group|acyclic group]]. This fact seems to be involved in most proofs of acyclicity of presently known acyclic groups [[#References|[a2]]]. Existentially closed groups are binate. Every group is a normal subgroup of a normal subgroup of a binate group.

The structure of binate groups is revealed by the study of a binate tower, i.e., a chain of groups $G_0\leq G_1\leq\dots$, homomorphisms $\alpha_i\colon G_i\to G_{i+1}$ and non-trivial structure elements $u_i\in G_{i+1}$ subject to relations of the above form. There is a universal binate tower $\mathcal U(G_0)$, obtained from $G_0$ by a sequence of HNN-extensions, with the characteristic property that every binate tower with base $G_0$ admits a unique smallest sub-binate tower, and this is a quotient binate tower of $\mathcal U(G_0)$ in a unique way [[#References|[a3]]]. In particular, the group $\mathcal U(1)$ is highly non-Hopfian and non-co-Hopfian (cf. also [[Hopf group|Hopf group]]). Its properties allow one to prove that binate groups admit no non-trivial finite-dimensional linear representation over any field [[#References|[a1]]], and similarly for many acyclic groups of automorphisms [[#References|[a5]]], [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Alperin,   A.J. Berrick,   "Linear representations of binate groups"  ''J. Pure Appl. Algebra'' , '''94'''  (1994)  pp. 17–23</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.J. Berrick,   "Universal groups, binate groups and acyclicity" , ''Proc. 1987 Singapore Group Theory Conf.'' , W. de Gruyter  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.J. Berrick,   K. Varadarajan,   "Binate towers of groups"  ''Arch. Math.'' , '''62'''  (1994)  pp. 97–111</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Varadarajan,   "Pseudo-mitotic groups"  ''J. Pure Appl. Algebra'' , '''37'''  (1985)  pp. 205–213</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.J. Berrick,   "Groups with no nontrivial linear representations"  ''Bull. Austral. Math. Soc.'' , '''50'''  (1994)  pp. 1–11</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.J. Berrick,   "Corrigenda: Groups with no nontrivial linear representations"  ''Bull. Austral. Math. Soc.'' , '''52'''  (1995)  pp. 345–346</TD></TR></table>