Stability results for Houghton groups (Q312410)

For a natural number \(n\in\mathbb N\), write \([n]=\{1,\dots,n\}\). The Houghton group \(\mathcal H_n\) is the group of all permutations \(g\) of the set \(\mathbb N\times[n]\) such that for each \(i\in [n]\) there is an integer \(d_i\in\mathbb Z\) with the property that \(g(x,i)=(x+d_i,i)\) for all \(x\) large enough. That is, for each element of \([n]\), the permutation \(g\) eventually acts by translation on \(\mathbb N\). The symmetric group \(\mathfrak S_n\) embeds into the set of all permutations of \(\mathbb N\times[n]\) by acting on \([n]\). Thus the twisted Houghton groups are the semidirect products \(\widetilde{\mathcal H}_n=\mathcal H_n\rtimes\mathfrak S_n\) for \(n\in\mathbb N\).NEWLINENEWLINEThe authors also introduce a multidimensional version of the (twisted) Houghton group \(\mathcal H_{k,n}\) (and \(\widetilde{\mathcal H}_{k,n}\)) as the groups of permutations of \(\mathbb N^k\times[n]\) that are translations on all rays of a finite partition of \(\mathbb N^k\times[n]\) for \(n\in\mathbb N\). In particular, \(\mathcal H_{1,n}=\mathcal H_n\) and \(\widetilde{\mathcal H}_{1,n}=\widetilde{\mathcal H}_n\) for all \(n\). Furthermore, there are inclusions \(\mathcal H_{k,n}\to\mathcal H_{k,n+1}\) and \(\widetilde{\mathcal H}_{k,n}\to\widetilde{\mathcal H}_{k,n+1}\) which induce maps in homology.NEWLINENEWLINEThe authors investigate the homology with trivial coefficients of the (twisted) Houghton groups and their multidimensional analogues. Their first main theorem states that the induced map in homology \(H_i(\widetilde{\mathcal H}_{k,n};\mathbb Z)\to H_i(\widetilde{\mathcal H}_{k,n+1};\mathbb Z)\) is surjective if \(i\leq\frac12(n-1)\) and injective if \(i\leq\frac12(n-2)\).NEWLINENEWLINEThe second main theorem in the article is a representation stability result asserting that, given a commutative noetherian ring \(R\), then for every \(i,k\in\mathbb N\) there exists an FI-module \(V\) given by \(V_n=H_i(\mathcal H_{k,n};R)\) and there exists a finitely generated FI-module \(W\) together with a map \(W\to V\) such that \(W_n\to V_n\) is surjective for all \(n\) large enough.