Verbal embeddings and a geometric approach to some group presentations (Q1822622)

For a ring R, the Steinberg group St(n,R) is the group generated by symbols \(x_{i,j}(r)\), \(1\leq i,j\leq n\), subject to the relations that \(x_{i,j}(r)x_{i,j}(s)=x_{i,j}(r+s)\), \([x_{i,j}(r),x_{j,k}(s)]=x_{i,k}(rs)\) for \(i\neq k\), and some commuting relations. The authors consider a generalization called a linkage group. A linkage graph \(\Gamma\) consists of a collection of vertices and a collection of ordered triples of vertices, called linkages. The linkage group L(R,\(\Gamma)\) is generated by symbols \(r_ v\), for all \(r\in R\) and \(v\in V(\Gamma)\) subject to the relations that \(r_ vs_ v=(r+s)_ v\) and that \([r_ u,s_ w]=(rs)_ v\) whenever (u,v,w) is a linkage. A (special) verbal embedding of R into a group G is a function \(\phi\) : \(R\to G\) such that \(\phi (r)\phi (s)=\phi (r+s)\) and the commutator \([a^{-1}\phi (r)a,b^{-1}\phi (s)b]=\phi (rs)\), where a and b are two fixed elements of G that generate a subgroup C of G, called the constant group. The authors prove that for every H with \(C\leq H\leq G\), there exists a linkage graph \(\Gamma_ H\) and an epimorphism from \(L(R,\Gamma_ H)\) to the subgroup of G generated by the H-conjugates of the subgroup generated by \(a^{-1}\phi (R)a\), \(\phi\) (R), and \(b^{- 1}\phi (R)b\). In particular, if G is simple, then taking \(H=G\) shows that G is a homomorphic image of a linkage group. The authors study linkage groups with the property that for each \(v\in V(\Gamma)\), the function sending r to \(r_ v\) is injective. Assuming that R has a (multiplicative) identity element and no zero divisors, this implies that for any linkage (u,v,w), the subgroup of G is isomorphic to the group U(R) of \(3\times 3\) upper triangular matrices with entries in R and all diagonal entries equal to 1. Moreover, it forces very strong restrictions on \(\Gamma\) ; in particular, it cannot contain certain kinds of subgraphs. This leads to various consequences; for example, C must have order at least 6, and if C is abelian then neither a nor b can be an involution. After a study of linkage graphs for the Steinberg groups (with \(n\geq 3)\), the authors present several other interesting examples. For instance, \(A_ 5\) does not admit a verbal embedding of a ring with identity and no zero divisors (this follows from the authors' theorem 3.1, not Theorem 2.1 as they have written), but for \(n\geq 6\), \(A_ n\) admits verbal embeddings of the ring with two elements. Other examples include verbal embeddings in the Mathieu groups, and in an infinite group. The final example is a group of order 160 which admits a verbal embedding of the ring \(\{\) 0,2,4,6\(\}\) of even integers modulo 8, wherein most of the strong results proved assuming that R has an identity element and no zero divisor fail to hold.