Invariable generation and the Houghton groups. (English) Zbl 1504.20004
A group \(G\) is invariably generated by a subset \(S\) if replacing each element of \(S\) with any of its conjugates results in a generating set for \(G\). The group \(G\) is invariably generated if such an \(S\) exists. Let us denote by \(d(G)\) the smallest cardinal of a generating set of \(G\). Let \(n\) be a positive integer and let \(X_n = \{1, \dots, n\} \times \mathbb{N}\). The Houghton group \(H_n\) is the group of permutations of \(X_n\) that eventually act as a translation in each copy of \(\mathbb{N}\).
Wiegold showed that \(H_3\) is invariably generated whereas its subgroup \(H_1\) isn’t [J. Wiegold, Arch. Math. 29, 571–573 (1977; Zbl 0382.20029)]. From now on, we assume that \(n \ge 2\).
The author first proves that \(H_n\) is invariably generated by a subset with \(d(H_n) = \max(2, n - 1)\) elements (Theorem A). Then he proves that every group commensurable to \(H_n\) is invariably generated by a finite subset (Theorem B).
Let us describe the invariable finite generating subsets constructed by the author. For \(k \in \{2, \dots ,n\}\), let \(g_k\) be the permutation of \(X_n\) with support \((\{1\} \times \mathbb{N}) \cup (\{k\} \times \mathbb{N})\) which is defined by \(g_k(1, m) = (1, m + 1)\) for \(m \ge 1\), \(g_k(k, 1) = (1 ,1)\) and \(g_k(k, m) = (k, m - 1)\) for \(m \ge 2\). The group \(H_2\) is invariably generated by the set \(\{ g_2, g_2 \sigma\}\) where \(\sigma\) is the \(4\)-cycle \(\left((1, 4)(1, 3)(1, 2)(1, 1)\right)\). If \(n \ge 3\), then \(H_n\) is invariably generated by \(\{g_2, \dots, g_{n - 1}, h\}\) where \(h = g_n g_{n - 1} \cdots g_2\).
The proof of Theorem B is the most delicate part of the paper. It relies on the following elementary fact: each subgroup \(H\) of finite index in \(H_n\) contains the subgroup generated by the \(3\)-cycles of \(X_n\) and the permutations \(g_2^{2v}, \dots, g_n^{2v}\) for some \(v = v(H) > 1\). The latter subgroup, named \(U_{2v}\), is easily seen to be normal and of finite index in \(H_n\). The author’s key finding is that \(U_{2v}\) is invariably generated by \(\{g_2^{2v}, \dots, g_{n - 1}^{2v}, h_{2v}, \sigma_1, g_2^{4v}\sigma_1, g_2^{4v} \sigma_2\}\) where \(h_{2v} = g_n^{2v} g_{n - 1}^{2v} \cdots g_2^{2v}\), \(\sigma_1\) is the \(3\)-cycle \(\left((1,1) (1,2) (1,3)\right)\) and \(\sigma_2\) is the product \(\left( (1, 1) (1,2)\right) \left( (1, 3) (1,4) \cdots (1, 4v)\right)\).
Wiegold showed that \(H_3\) is invariably generated whereas its subgroup \(H_1\) isn’t [J. Wiegold, Arch. Math. 29, 571–573 (1977; Zbl 0382.20029)]. From now on, we assume that \(n \ge 2\).
The author first proves that \(H_n\) is invariably generated by a subset with \(d(H_n) = \max(2, n - 1)\) elements (Theorem A). Then he proves that every group commensurable to \(H_n\) is invariably generated by a finite subset (Theorem B).
Let us describe the invariable finite generating subsets constructed by the author. For \(k \in \{2, \dots ,n\}\), let \(g_k\) be the permutation of \(X_n\) with support \((\{1\} \times \mathbb{N}) \cup (\{k\} \times \mathbb{N})\) which is defined by \(g_k(1, m) = (1, m + 1)\) for \(m \ge 1\), \(g_k(k, 1) = (1 ,1)\) and \(g_k(k, m) = (k, m - 1)\) for \(m \ge 2\). The group \(H_2\) is invariably generated by the set \(\{ g_2, g_2 \sigma\}\) where \(\sigma\) is the \(4\)-cycle \(\left((1, 4)(1, 3)(1, 2)(1, 1)\right)\). If \(n \ge 3\), then \(H_n\) is invariably generated by \(\{g_2, \dots, g_{n - 1}, h\}\) where \(h = g_n g_{n - 1} \cdots g_2\).
The proof of Theorem B is the most delicate part of the paper. It relies on the following elementary fact: each subgroup \(H\) of finite index in \(H_n\) contains the subgroup generated by the \(3\)-cycles of \(X_n\) and the permutations \(g_2^{2v}, \dots, g_n^{2v}\) for some \(v = v(H) > 1\). The latter subgroup, named \(U_{2v}\), is easily seen to be normal and of finite index in \(H_n\). The author’s key finding is that \(U_{2v}\) is invariably generated by \(\{g_2^{2v}, \dots, g_{n - 1}^{2v}, h_{2v}, \sigma_1, g_2^{4v}\sigma_1, g_2^{4v} \sigma_2\}\) where \(h_{2v} = g_n^{2v} g_{n - 1}^{2v} \cdots g_2^{2v}\), \(\sigma_1\) is the \(3\)-cycle \(\left((1,1) (1,2) (1,3)\right)\) and \(\sigma_2\) is the product \(\left( (1, 1) (1,2)\right) \left( (1, 3) (1,4) \cdots (1, 4v)\right)\).
Reviewer: Luc Guyot (Genève)
MSC:
| 20B22 | Multiply transitive infinite groups |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
invariable generation of infinite groups; finite index subgroups; commensurable groups; generators; Houghton groupsCitations:
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