2. GR- and GR $^*$ -group

A system of equations over a group is called nonsingular if the integer rows composed of the exponent-sums of unknowns in equations are linearly independent. For example, the system (with unknowns $x,y,z,t$ over a group $G \mathrel\backepsilon a,b,c,d$ ):

\begin{equation*}\left\{ \begin{array}{r@{\,}c@{\,}l}axbycy{z^5}d{z^{ - 2}} &=& 1\\ \\[-8pt]{[xt,dz]_{}}d{x^4}c{y^5}b{z^6} &=& 1\\ \\[-8pt]a{x^7}{y^8}d{z^k} &=& 1\end{array} \right.\,\,\textrm{has}\,\textrm{the}\,\textrm{exponent} - \textrm{sum}\,\textrm{matrix}\,\left( {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1&2&3&0\\ \\[-8pt]4&5&6&0\\ \\[-8pt]7&8&k&0\end{array}} \right),\end{equation*}

i.e., this system is singular if and only if $k = 9$ .

Gerstenhaber–Rothaus theorem [Reference Gerstenhaber and Rothaus12]. Any nonsingular system of equations over a finite group is solvable over this group.

We suggest to call a group G a GR-group, if any nonsingular system of equations over G is solvable over G. Pestov [Reference Pestov33] noticed that all hyperlinear group are GR-groups.

It is easy to see that

\begin{equation*}the\,class\,of\,GR\hbox{-}groups\,is\,a\,quasivariety,\end{equation*}

i.e., this class consists of all groups satisfying some (possibly, infinite) system of quasi-identities, i.e., (finite) formulae of the form:

\begin{equation*}(\forall x,y, \ldots )({u_1}(x,y, \ldots ) = 1\;\& \ldots \;\& \;{u_k}(x,y, \ldots ) = 1 \Rightarrow v(x,y, \ldots ) = 1),\end{equation*}

where ${u_i}(x,y, \ldots )$ and $v(x,y, \ldots )$ are words in the alphabet $\{ {x^{ \pm 1}},{y^{ \pm 1}}, \ldots \} $ .

A simple way to prove this is to apply the following characterisation of quasivarieties [Reference Mal’cev31]:

\begin{equation*}\begin{array}{l}a\,nonempty\,class\,of\,groups\,is\,a\,quasivariety\,if\,and\,only\,if\,it\,is\,closed\\under\,passage\,to\,subgroups\,and\,reduced\, (= filtered)\,products.\end{array}\end{equation*}

For the class of GR-groups, the both conditions hold obviously.

A disadvantage of the quasivariety of GR-groups is that it is unclear whether this class is closed under free products. We call a group G a GR $^*$ -group if it satisfies the following equivalent conditions:

1. (1) the free product $G*\mathbb{Z}$ is a GR-group;

2. (2) the free product $G*G$ is a GR-group;

3. (2’) the free product of any family of groups isomorphic to G is a GR-group;

4. (4) G embeds into a GR-group $\tilde G \supseteq G$ satisfying no nontrivial mixed identity with constants from G, i.e., for any word $v(x,y, \ldots ) \in G*F(x,y, \ldots )\backslash \{ 1\} $ (where $F(x,y, \ldots )$ is a free group), there exist ${\tilde x_v},{\tilde y_v}, \ldots \in \tilde G$ such that $v(\tilde x,\tilde y, \ldots ) \ne 1$ .

Proposition. These conditions are indeed equivalent.

Proof. First, note that, if $|G| \le 2$ , then all the conditions are equivalent, because they are true in this case (the class of residually finite groups is closed with respect to free products [Reference Gruenberg13] and is contained in the class of GR-groups by the Gerstenhaber–Rothaus theorem, therefore, if $|G| = 2$ , then we can take, e.g., the free product of G and the free group of countable rank as $\tilde G$ in (3)).

If $|G| \gt 2$ , then (1) and (2) are equivalent, because $G*\mathbb{Z}$ and $G*G$ embed into each otherFootnote 1 (and any subgroup of a GR-group is a GR-group).

Condition (2’) is equivalent to (1) and (2) by similar reasons:

• – the free product of any family of groups isomorphic to G is residually a finite free product $G* \ldots *G$ (because any element of an arbitrary free product lies in a finite subproduct, and each subproduct is a retract of the whole product);

• – GR is a residual property (because the class of GR-groups is a quasi-variety);

• – a finite free product $G* \ldots *G$ (with at least two factors) and $G*\mathbb{Z}$ embed into each other by the same ‘converse of the Kurosh theorem’Footnote 1.

Moreover, Condition (1) implies Condition (3), because $G*\mathbb{Z}$ contains the free product $\tilde G = G*{F_\infty }$ of G and the free group of infinite (countable) rank, which, obviously, has no mixed identities with constants from G.

It remains to prove the implication (3) ⇒ (1). The Cartesian product of copies ${\tilde G_v}$ of $\tilde G$ is a GR-group (because the Cartesian product of any family of GR-groups is, obviously, a GR-group). The group $G*\langle t\rangle_\infty $ embeds into H as follows: G embeds diagonally, and the element t is mapped to the element whose vth coordinate is ${\tilde t_v} \in \tilde G$ (i.e. $v({\tilde t_v}) \ne 1$ ). Clearly, this is an embedding. It remains to recall that a subgroup of a GR-group is a GR-group too. This completes the proof.

Question 1. Are the classes of GR- and GR $^*$ -groups closed with respect to direct and free products?

Simple considerations show that

• – the class of GR-groups is, obviously, closed with respect to direct products;

• – if the class of GR-groups is closed under free products, then GR-groups and GR $^*$ -groups are the same (this follows immediately from Condition 2) of the definition of GR $^*$ -groups; more generally, any GR-group decomposable nontrivially into a free product is a GR $^*$ -group);

• – if the class of GR $^*$ -groups is closed under direct products or, more generally, if there exists a GR^*-group containing any two given GR^*-groups as subgroups, then the class of GR $^*$ -groups is closed under free products.

This means that the answers to Question 1 can be only the following:

• – either GR = GR $^*$ , and this class is closed with respect to both operations,

• – or the class of GR-groups is closed under direct products, but not closed under free products, and the class of GR $^*$ -groups is

  • – either closed under neither operations,

  • – or closed with respect to both operations,

  • – or closed under free products, but not closed under direct products.

Certainly, none of the presently known facts contradicts the equalities GR = GR^*={all groups} (Howie’s conjecture). The class of hyperlinear groups is closed under free products [Reference Brown, Dykema and Jung5]; hence, all hyperlinear groups are not only GR- but also GR $^*$ -groups.

3. Main result

Suppose that $A \triangleleft G$ and F is a free group. We call the natural epimorphism ${\varepsilon _A} \colon G*F \to (G/A)*F$ the $G/A$ -content; the $G/G$ -content is just the content $\varepsilon $ (defined in [Reference Klyachko and Thom25, Reference Nitsche and Thom32]).

Main theorem. Suppose that a group G contains a normal subgroup A, which is GR-group, and the quotient group $G/A$ is locally indicable. Then an equation $w(x,y, \ldots ) = 1$ is solvable over G if the $G/A$ -content of $w(x,y, \ldots )$ is not conjugate to an element of $G/A$ in $(G/A)*F(x,y, \ldots )$ .

This theorem implies immediately the Freiheitssatz stated in the introduction. Indeed, consider the group $G = C*D*K$ , its normal subgroup $A = \left\langle {\left\langle K \right\rangle } \right\rangle $ (which is GR-group, because it is isomorphic to the free product of a family of groups isomorphic to the GR $^*$ -group K), and the embedding $\varphi\,{:}\,G \to G*{\left\langle {{t_{}}} \right\rangle _\infty }$ , where $C \mathrel\backepsilon c \mapsto {c^t} \in G*{\left\langle t \right\rangle _\infty }$ and the other free factors are mapped identically. The equation $\phi (w) = 1$ (with one variable t) is solvable over G by the main theorem, i.e., the composition ${G}\mathrel{\mathop{\longrightarrow}\limits^{\varphi}}G*{\left\langle t \right\rangle _\infty } \to G*{\left\langle t \right\rangle _\infty }/\left\langle {\left\langle {\phi (w)} \right\rangle } \right\rangle $ is injective on $D*K$ and contains w in its kernel, as required. The injectivity of the natural mapping of $C*K$ can be proven similarly.

4. Key lemma

A proof of the following result (obtained essentially in [Reference Nitsche and Thom32]) can be found in the last section.

Nitsche–Thom theorem. A system of equations $\{ {w_1}(x,y, \ldots ) = 1,\;{w_2}(x,y, \ldots ) = 1, \ldots \} $ over a GR-group G is solvable over G if the standard complex of the presentation $\langle x,y,\dots\;|\; \varepsilon(w_1)=1,\,\varepsilon(w_2)=1,\dots\rangle $ , where $\varepsilon $ stands for the content, admits a covering with trivial second homologies over $\mathbb{Z}$ .

To apply this theorem, we need the following lemma.

Lemma. Suppose that a locally indicable group L acts freely on a set X, and $L\times F(X)\mathrel{\mathop{\longrightarrow}\limits^{\circ}} F(X)$ is the natural extension of this action to the free group with basis X. Then, for each word $v \in F(X)\backslash \{ 1\} $ , some covering of the standard complex K of the presentation $H = \left\langle {X\;|\;L \circ v} \right\rangle $ has trivial second homologies.

Proof. Suppose that $v = {u^k}$ and the word u is not a proper power in $F(X)$ . Let us prove that the required covering is the covering $p\colon\tilde K \to K$ corresponding to the normal closure $\left\langle {\left\langle {L \circ u} \right\rangle } \right\rangle \triangleleft H = {\pi _1}(K)$ of the orbit of u. In an explicit form, the complex $\tilde K$ is the Cayley graph of the group $A = \left\langle {X\;|\;L \circ v} \right\rangle $ with ‘k times wrapped’ two-discs glued to each cycle with label $l \circ u$ , where $l \in L$ (so, the words $l \circ v = l \circ {u^k}$ are written on the boundaries of the discs).

The group $A = \left\langle {X|L \circ v} \right\rangle $ embeds into $\hat A = Y*L/\left\langle {\left\langle {\hat u} \right\rangle } \right\rangle $ , where Y is a set of representatives of orbits of the action of L on X, and the word $\hat u$ is obtained from u by replacing each letter x with $ly{l^{ - 1}}$ , where $l \in L$ and $y \in Y$ are the (only) elements such that $x = l \circ y$ . The embedding $A \to \hat A$ is obvious: $x = l \circ y \mapsto ly{l^{ - 1}}$ . The group $\hat A$ is locally indicable [Reference Brodskii4] and $\hat A = L\rightthreetimes A$ . This group $\hat A$ acts on the complex $\tilde K$ : the action of A is the standard action of a group on its Cayley graph, and L acts by conjugation on vertices: $l \circ a = la{l^{ - 1}}$ (then vertices joined by an edge labelled x are mapped to vertices joined by an edge labelled $l \circ x$ ). This action of $\hat A$ is transitive on two-cells and on vertices and free on (oriented) edges.

The first is quite obvious; and the second can be explained easily: the stabiliser of an edge must stabilise its origin; the action is vertex-transitive; therefore, it suffices to show that the stabiliser of each edge e outgoing from the identity is trivial; so, the equality $(l,a) \circ e = e$ implies that $a = 1$ (otherwise, we would obtain an edge with another origin); now, the equality $l \circ e = e$ means that $l = 1$ , because the action of L on X is free.

Now, the situation is simple. Since the action on two-cells is transitive, the nontriviality of the second homologies implies an equality $r\bar u = 0$ , where

• – $\bar u$ is the sum of edges of the boundary of a two-cell (which does not vanish, because the universal covering of a group with one relator $u = 1$ , which is not a proper power, is contractible, i.e., one-relator torsion-free groups are aspherical, see [Reference Lyndon and Schupp29]),

• – and r is a nonzero element of the group ring $\mathbb{Z} [\hat A]$ of $\hat A$ .

The freeness of the action on edges leads immediately to zero divisors in the group ring $\mathbb{Z}[\hat A]$ of the locally indicable group $\hat A$ , which is a contradiction [Reference Higman14].

5. Proof of the main theorem

Since the word ${\varepsilon _A}(w) \in (G/A)*F(x,y, \ldots )$ is not conjugate to an element of $G/A$ , the equation ${\varepsilon _A}(w) = 1$ has a solution in a locally indicable group B containing $G/A$ (by the Brodskii–Howie–Short theorem). Therefore, we have the embeddings of the G into the unrestricted wreath products: $G \subseteq A\bar{\wr} (G/A) \subseteq A \bar\wr B = {A^B}\leftthreetimes B$ (where the first embedding is the Kaloujnine–Krasner theorem [Reference Krasner and Kaloujnine26], see also [Reference Kargapolov and Merzljakov19]). Therefore, replacing G with $A\bar\wr B$ , and A with its Cartesian power ${A^B} \triangleleft A\bar{\wr} B$ (which is also a GR-group), we can assume that the equation ${\varepsilon _A}(w) = 1$ has a solution $\hat x,\hat y, \ldots \in G/A$ ; hence, making an obvious change of variables ( $x \mapsto x\hat x,\;y \mapsto y\hat y, \ldots $ ), we obtain that ${\varepsilon _A}(w)$ is contained in the normal closure $\left\langle {\left\langle F \right\rangle } \right\rangle $ of $F = F(x,y, \ldots )$ in $(G/A)*F$ . Thus, we assume that $w = w(x,y, \ldots )$ can be rewritten as a word $\hat w$ in the alphabet $\{ {x^{ \pm b}},{y^{ \pm b}}, \ldots \;|\;b \in B = G/A\} \sqcup A$ (because we assume already that $G = A\leftthreetimes B$ ).

We arrive to the situation of the Lemma. For $X = \{ bx{b^{ - 1}},by{b^{ - 1}}, \ldots \;|\;b \in B\} $ and $v = \varepsilon (\hat w) \in F(X)$ , where $\varepsilon \colon F(X)*A \to F(X)$ is the natural retraction, and $L = B$ , the Lemma says that a covering of the standard complex of the presentation $H = \left\langle {X\;|\;B \circ \varepsilon (\hat w)} \right\rangle $ has trivial second homologies. By the Nitsche–Thom theorem, this means the solvability of the system of equations $\{ b \circ w = 1\;|\;b \in B\} $ (with unknowns $X = B \circ \{ x,y, \ldots \} $ ) over the group A (where the action of B on X is extended to $A*F(X)$ naturally: $b \circ a\, :\!=\, ba{b^{ - 1}}$ for $a \in A$ ). In other words, the natural mapping $A \to \tilde A = (A*F(X))/\left\langle {\left\langle {\{ b \circ w = 1\;|\;b \in B\} } \right\rangle } \right\rangle $ is injective. The group B acts on $\tilde A$ , and this action extends the action of B on A by conjugations. Therefore, the natural mapping $G = A\leftthreetimes B \to \tilde G = \tilde A\leftthreetimes B$ is injective. Moreover, $w(x,y, \ldots ) = 1$ in $\tilde G$ , which means the solvability of the equation.

6. Subgroup presentations and the Nitsche–Thom theorem

In this section, we prove the Nitsche–Thom theorem as stated in Section 4. This result was essentially proven in [Reference Nitsche and Thom32], but unfortunately, a weaker version was explicitly stated there (see [Reference Nitsche and Thom32], Theorem 1.3 and Remark 2.2). Our approach somewhat differs from the proof that can be extracted from [Reference Nitsche and Thom32].

The following fact is well known, see, e.g., [Reference Zieschang, Vogt and Coldewey37], Theorem 2.2.1, or [Reference Lyndon and Schupp29], Proposition II.4.1.

Schreier’s theorem on subgroup presentations [Reference Schreier34]. Suppose that U is a subgroup of a group $G = \left\langle {X|R} \right\rangle $ , and $T \subseteq F(X)$ is a Schreier system of the right coset representatives of ${\pi ^{ - 1}}(U)$ in the free group $F(X)$ , where $\pi\,{:}\, F(X) \to G = F/\left\langle {\left\langle R \right\rangle } \right\rangle $ is the natural epimorphism. Then the subgroup U is generated by the images of the words ${y_{t,x}} = tx{(\overline{tx})^{ - 1}} \in F(X)$ and has a presentation, where generators are all nontrivial words $\{ {y_{t,x}}\;|\;t \in T,x \in X\} $ , and the defining relators are all words $\{ tr{t^{ - 1}}\;|\;t \in T,x \in X\} $ rewritten as words in nontrivial generators ${y_{t,x}}$ considered as letters.

Here, the bar means taking a representative: $\bar v \in T$ (where $v \in F(X)$ ) is the unique word from T such that ${\pi ^{ - 1}}(U)\bar v = {\pi ^{ - 1}}(U)v$ ; and T is Schreier in the usual sense: any prefix of any word from T lies in T.

The geometric interpretation of this presentation, which we call Schreier, is also well known:

• – the group G is the fundamental group of the standard complex K corresponding to the presentation $\left\langle {X|R} \right\rangle $ (i.e., K has a single vertex, edges correspond to generators from X, and two-dimensional cells correspond to relators from R);

• – the subgroup U is the fundamental group of the complex $\tilde K$ (with a base vertex) covering the complex K; vertices of $\tilde K$ correspond to right cosets of U in G (and of ${\pi ^{ - 1}}(U)$ in $F(X)$ );

• – a Schreier system of representatives corresponds to a maximal subtree in the 1-skeleton of $\tilde K$ ;

• – generators ${y_{t,x}}$ correspond to edges of $\tilde K$ (or, more precisely, each edge $e \in \tilde K$ corresponds to a path starting at the base vertex going through the maximal subtree to the start point of e, passing e, and returning to the base vertex via the maximal subtree); so, nontrivial generators ${y_{t,x}}$ correspond to edges not belonging to the maximal subtree;

• – finally, relators of U correspond to two-cells of $\tilde K$ .

This geometric interpretation shows that

Thus, we can restate the Nitsche–Thom theorem in group-theoretical terms (with no topology or homologies).

Nitsche–Thom theorem (a pure group-theoretical form). A system of equations $W = 1$ (possibly, infinite and, possibly, with an infinite set of unknowns X) over a GR-group G is solvable over G if there exists a subgroup of $\left\langle {X|\varepsilon (W)} \right\rangle $ , the relators of whose Schreier presentation form a nonsingular system. (Here, $\varepsilon $ stands for the content, see Introduction.)

Proof. Suppose that $A \subseteq \left\langle {X|\varepsilon (W)} \right\rangle $ is a subgroup with a nonsingular Schreier presentation. Consider a presentation $G = \left\langle {Z|V} \right\rangle $ of the group G, the corresponding presentation of the group $(G*F(X))/\left\langle {\left\langle W \right\rangle } \right\rangle = \left\langle {Z \sqcup X|V \sqcup W} \right\rangle = H$ , the natural epimorphism $\theta\,{:} \,H \to H/\left\langle {\left\langle G \right\rangle } \right\rangle = \left\langle {X|\varepsilon (W)} \right\rangle $ , and the subgroup ${\theta ^{ - 1}}(A) \subseteq H$ . It is easy to see that the Schreier presentation for the subgroup ${\theta ^{ - 1}}(A) \subseteq H$ transforms into the Schreier presentation for the subgroup $A \subseteq \left\langle {X|\varepsilon (W)} \right\rangle $ by deleting all generators corresponding to conjugate of generators from Z (in geometrical terms, each vertex of the covering complex corresponding to the subgroup ${\theta ^{ - 1}}(A) \subseteq H$ is contained in a subcomplex isomorphic to the standard complex of the presentation $G = \left\langle {Z|V} \right\rangle $ ; and, contracting each such subcomplex to a single point, we obtain the covering complex corresponding to the subgroup $A \subseteq \left\langle {X|\varepsilon (W)} \right\rangle $ ).

Thus, the relators of the group ${\theta ^{ - 1}}(A)$ form a nonsingular system of equations over the free product of $|\!\left\langle {X|\varepsilon (W)} \right\rangle \,{:}\,A|$ copies of G. Since G is a GR-group, the natural mapping $G \to {\theta ^{ - 1}}(A) \subseteq H$ is injectiveFootnote 2 (to show this, we can take the quotient group of ${\theta ^{ - 1}}(A)$ by the normal closure of the product of all copies of G, except one). This implies immediately the injectivity of the natural homomorphism $G \to H$ , as required.