Let \(G\) be a word hyperbolic group and \(H\) a subgroup of \(G\). The author defines \(H\) to be a Gromov subgroup or more briefly a \(G\)-subgroup if \(H\) is non-elementary (that is, \(H\) is not a finite extension of a cyclic group) and if for any finite subset \(M\subset G\) there is a non-elementary hyperbolic quotient \(G_1\) of \(G\) such that the natural map \(G\to G_1\) is surjective on \(H\) and injective on \(M\). As the author points out, there is an erroneous statement in the paper Hyperbolic groups [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] of M. Gromov which claims that any non-elementary subgroup of a hyperbolic group is a \(G\)-subgroup. The author presents as a counter-example a group he had already defined in 1989, and another example which shows that the result in question is false even if one does not require injectivity on a given finite subset \(M\). The goal of the paper is in fact to present different results on subgroups of hyperbolic groups which are related to these matters.
The main results are the following. Let \(G\) be a non-elementary hyperbolic group. Then: Theorem 1. Let \(H\) be a non-elementary subgroup of \(G\) and \(E(H)\) the set of elements \(x\in G\) such that the orbit of \(x\) under the action of \(H\) on \(G\) by conjugation is finite. Then \(H\) is a \(G\)-subgroup if and only if \(E(H)=E(G)\) and for every \(g\in G\), there exists an element \(h\in H\) with \(gag^{-1}=hah^{-1}\) for every \(a\in E(G)\).
Theorem 2. Let \(H_1,\dots,H_k\) be \(G\)-subgroups of \(G\) and let \(H_1',\dots,H_{k'}'\) be non-elementary subgroups of \(G\). Then, for any finite subset \(M\subset G\), there is a quotient \(G_1\) of \(G\) such that: (1) \(G_1\) is a non-elementary hyperbolic group; (2) the natural homomorphism \(\varepsilon_1:G\to G_1\) is injective on \(M\) and surjective on each subgroup \(H_1,\dots,H_k\); (3) \(\varepsilon_1\)-images of elements from \(M\) are conjugate in \(G_1\) if and only if the elements are conjugate in \(G\); (4) the centralizer \(C_{G_1}(\varepsilon_1(a))\) for every \(a\in M\) is the \(\varepsilon_1\)- image of the centralizer \(C_G(a)\); (5) \(\text{Ker }\varepsilon_1\) is a torsion free subgroup; (6) \(\varepsilon_1\) induces a bijective map on sets of conjugacy classes of elements of finite orders in \(G\) and \(G_1\) respectively; (7) \(\varepsilon_1\)-images of the subgroups \(H_1',\dots,H_{k'}'\) are non-elementary.
Theorem 3. Let \(E_0\) be a maximal elementary subgroup of \(G\), let \(C\) be an infinite cyclic normal subgroup of \(E_0\) and \(M\) a finite subset of \(G\). Then there exists an integer \(m_0=m_0(G,E_0,M)\) such that for any \(m\geq m_0\) there is a quotient \(G_1\) of \(G\) such that (1) \(G_1\) is a non-elementary hyperbolic group; (2) the natural homomorphism \(\varepsilon_1:G\to G_1\) is injective on \(M\); (3) the image \(\varepsilon_1(E_0)\) is isomorphic to the quotient \(E_0/C^m\); (4) every element of finite order in \(G_1\) is the image of an element having finite order in \(G\), or it is conjugate to an element of \(\varepsilon_1(E_0)\); (5) \(\text{Ker }\varepsilon_1\) is torsion free; (6) if \(m\) is even or \(G\) has no non-central elements of order \(2^k\), then the \(\varepsilon_1\)-images of elements of \(M\) are conjugate in \(G_1\) if and only if they are conjugate in \(G\); (7) if \(G\) has non-central elements of order \(2^k\), then for every \(a\in M\), the centralizer \(C_{G_1}(\varepsilon(a))\) is the \(\varepsilon_1\)-image of the centralizer \(C_G(a)\).
Theorem 4. Suppose that \(G\) satisfies the following quasi-identical relation: \((\forall x,y\in G)\;x^2y=yx^2\Rightarrow xy=yx\). Suppose that every non-elementary subgroup of \(G\) is a \(G\)-subgroup and moreover, the hypotheses of Theorem 2 (resp. Theorem 3 for odd \(m\geq m_0)\) hold. Then one can add to the statement of Theorem 2 (resp. Theorem 3) that \(G_1\) satisfies the quasi-identical relation, that \(E(G_1)\) is a natural image of \(E(G)\) and all non-elementary subgroups of \(G_1\) are \(G\)-subgroups. Moreover, all abelian subgroups of \(G_1\) are cyclic if the same condition holds for \(G\).