Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers. (Q2873305)

Let \(\Gamma\) be a group. A group \(G\) is said be fully residually \(\Gamma\) if for every finite set of non-trivial elements \(\{g_1,g_2,\ldots,g_n\}\) of \(G\) there exists a homomorphism \(\varphi\colon G\to\Gamma\) such that \(\varphi(g_i)\neq 1\) for \(i=1,2,\ldots,n\). If this condition is only required to hold for \(n=1\), then the group is residually \(\Gamma\).NEWLINENEWLINE Let \(\Gamma\) be a group generated by the finite set \(A\) and \(F(X)\) be the free group on the finite set \(X\) of free generators. Let \(\Gamma[X]=\Gamma*F(X)\). A subset \(S=S(X,A)\) of \(\Gamma[X]\) is a system of equations over \(\Gamma\) and a solution of this system is a homomorphism \(\varphi\) from \(\Gamma[X]\) onto \(\Gamma\) which is identical on \(\Gamma\) and \((s)\varphi=1\) for every \(s\in S\). If \(\text{ncl}(S)\) denotes the normal closure of \(S\) in \(\Gamma[X]\), then every solution to the system \(S\) factors through the quotient \(\Gamma_S=\Gamma[X]/\text{ncl}(S)\). The radical of \(S\) over \(\Gamma\) is defined by \(R_\Gamma(S)=\{t\in\Gamma[X]\) such that if \(\varphi\) is a solution of the system \(S\), then \((t)\varphi=1\}\). The radical is a normal subgroup and the quotient group \(\Gamma_{R_\Gamma(S)}=\Gamma[X]/R_\Gamma(S)\) is the coordinate group of \(S\) over \(\Gamma\).NEWLINENEWLINE In this paper the authors, using equations over torsion free hyperbolic groups, study embeddings of residually hyperbolic groups into ``appropriate'' groups.NEWLINENEWLINE We quote some of their results which are used for the proof of their main result, but are interesting in their own.NEWLINENEWLINE Lemma 1. (Lemma 1.2) Let \(S(X)\) be a system of equations over a group \(G\). If \(G_S\) is residually \(G\), then \(R_G(S)=\text{nlc}(S)\) and hence \(G_{R(S)}=G_S\).NEWLINENEWLINE A special type of systems of equations over a group \(G\) is defined (see paragraph 3.1 in the paper) called NTQ systems. In the sequel a finitely generated torsion free hyperbolic group \(\Gamma=\langle A\mid\mathcal R\rangle\) is fixed.NEWLINENEWLINE Proposition. (Corollary 3.12) Let \(S(X,A)\) be an embeddable NTQ system over a toral relatively hyperbolic group \(\Gamma\). Then \(\Gamma_S\) embeds into a group obtained from \(\Gamma\) by extensions of centralizers. In particular, \(\Gamma_S\) is fully residually \(\Gamma\), is toral relatively hyperbolic, and \(\Gamma_S=\Gamma_{R(S)}\). -- Further, the embedding may be computed effectively.NEWLINENEWLINE Lemma 2. (Lemma 3.15) There is an algorithm that, given a finite presentation \(\langle Z\mid S\rangle\) of a group \(G\), produces (i) finitely many fully residually \(\Gamma\) groups \(G_1,\ldots,G_m\). (ii) homomorphisms \(\alpha_i\colon G\to G_i\), such that for every homomorphism \(\psi\colon G\to\Gamma\) there exists a homomorphism \(\widehat\varphi\colon G_i\to\Gamma\) such that \(\psi=\alpha_i\widehat\varphi\). Further, each group \(G_i\) has the form \(G_i=\Gamma_{T_i}\) where \(T_i\) is an embeddable \(\Gamma\)-NTQ system and the homomorphism \(\widehat\varphi\) is a \(\Gamma\)-homomorphism.NEWLINENEWLINE Corollary 1. (Corollary 3.17) If \(G\) is fully residually \(\Gamma\), then there exists \(i\in\{1,\ldots,m\}\) such that the homomorphism \(\alpha_i\) is injective. If \(G\) is residually \(\Gamma\), then for every \(g\in G\) there exists \(i\in\{1,\ldots,m\}\) such that \((g)\alpha_i\neq 1\).NEWLINENEWLINE Corollary 2. (Corollary 3.19) The universal theory of a torsion free hyperbolic group is decidable.NEWLINENEWLINE Theorem. (Main result (Theorem 3.21.)) Let \(\Gamma\) be any torsion free hyperbolic group. There is an algorithm that, given a finitely presented group \(G\), constructs (i) finitely many groups \(H_1,\ldots,H_n\), each given as a series of extensions of centralizers of \(\Gamma\), (ii) homomorphisms \(\varphi_i\colon G\to H_i\), such that (1) if \(G\) is fully residually \(\Gamma\), then at least one of the \(\varphi_i\) is injective, (2) if \(G\) is residually \(\Gamma\), the map \(\varphi_1\times\cdots\times\varphi_n\colon G\to H_1\times\cdots\times H_n\) is injective. -- This also holds for \(G\) in the category of \(\Gamma\)-groups.NEWLINENEWLINE As a corollary it is obtained a polynomial-time solution to the word problem in any finitely presented residually \(\Gamma\)-group.