Simple \(p\)-adic Lie groups with abelian Lie algebras (Q6566117)

I describe the overview of this paper extracted from the abstract and the introduction.\N\N``For each prime \(p\) and each positive integer \(d\), we construct the first examples of second countable, topologically simple \(p\)-adic Lie groups of dimension \(d\) whose Lie algebras are abelian. This answers several questions of Glöckner and Caprace-Monod. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.''\N\NOne of the main results of the authors is Theorem 1.2.\N\NTheorem 1.2. For each prime \(p\) and every positive integer \(d\), there exists a continuum of abstract isomorphism classes of second countable, topologically simple \(p\)-adic Lie groups of dimension \(d\) whose Lie algebras are abelian.\N\NAs an immediate consequence of Theorem 1.2, Corollary 1.3 follows as an affirmative answer.\N\NCorollary 1.3. Extraordinary \(p\)-adic Lie groups of dimension \(d\) do exist for every \(d > 1\) and every prime \(p\).\N\NThe authors' construction yields examples of topologically simple, totally disconnected, locally compact, second countable groups with compact open subgroups isomorphic to direct products of groups \(\mathbb{Z}_p\) for different primes (see Theorem 4.6).\N\NThe proof of the authors' main result employs geometric tools from the theory of group actions on hyperbolic spaces.\N\NThe groups from Theorem 1.2 arise as closed subgroups of the automorphism group of some countable discrete groups \(G_{\infty}\), themselves defined as unions of ascending chains \((G_n)\) of finitely generated groups with suitable properties (see Proposition 4.4). The construction of the groups \(G_n\) makes use of a generalization of the small cancellation technique developed in [\textit{M. Hull}, Groups Geom. Dyn. 10, No. 4, 1077--1119 (2016; Zbl 1395.20028); \textit{A. Yu. Ol'shanskij}, Int. J. Algebra Comput. 3, No. 4, 365--409 (1993; Zbl 0830.20053); \textit{D. Osin}, Ann. Math. (2) 172, No. 1, 1--39 (2010; Zbl 1203.20031)].\N\NIn Section 3 the authors develop a method for constructing words with small cancellations in weakly relatively hyperbolic groups and apply it to central extensions of acylindrically hyperbolic groups. I suggest referring to [\textit{I. Chifan} et al., ``Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces'', Preprint, \url{arXiv:2304.07455}].\N\NProposition 4.4. For any countable group \(R\) and arbitrary integers \(d, p > 1\), there exists an infinite ascending chain of groups \(G_1\le G_2\le \cdots\) such that the following conditions hold for all \(n \in \mathbb{N}\). \N\begin{itemize}\N\item[(i)] \(G_n\) is finitely generated and perfect.\N\item[(ii)] \(G_n / Z(G_n)\) is simple and there is an isomorphism \(\Psi_n :Z(G_n) \rightarrow Z^d\).\N\item[(iii)] \(Z(G_{n+1})\le Z(G_n)\) and \(\Psi_n(Z(G_{n+1}))=pZ^d\).\N\item[(iv)] \(C_{G_n}(G_1)=Z(G_1)\).\N\item[(v)] \(R\) is a subgroup of \(G_1\) , with \(R \cap Z(G_1)=\)\{1\}.\N\item[(vi)] If \(R\) is torsion-free, then so is \(G_n\). \N\end{itemize}\NIf in addition \(p > 2\), then the countable group \(G_{\infty}=\bigcup^{\infty}_{n=1}G_n\) is simple.\N\NTheorem 4.6. Suppose that \(d > 1\) and \(p > 2\) are any integers. Let \(R\) be any countable group and let \(G_1\le G_2\le \cdots\) be the ascending chain of groups provided by Proposition 4.4. Let also \(G_{\infty}=\bigcup_{n \in \mathbf{N}} G_n\). Then the topological group \(\mathcal{G}=\mathrm{Linn}(G_{\infty})=\overline{\mathrm{Inn}(G_{\infty})}\) has the following properties.\N\begin{itemize}\N\item[(i)] \(\mathcal{G}\) is non-abelian, locally compact, totally disconnected, non-discrete and topologically simple.\N\item[(ii)] \(\mathcal{G}\) is not abstractly simple; every non-trivial normal subgroup of \(\mathcal{G}\) contains Inn\((G_{\infty})\).\N\item[(iii)] The closure of the image of \(Z(G_1)\) in \(\mathcal{G}\) is a compact open subgroup isomorphic to \(Z_{p_1}^d\times \cdots \times Z_{p_t}^d\), where \(p_1 < p_2 < \cdots < p_t\) denote the distinct prime divisors of \(p\).\N\item[(iv)] \(R\) is embedded as a discrete subgroup in \(\mathcal{G}\).\N\item[(v)] \(\mathcal{G}\) is second countable, but not compactly generated.\N\item[(vi)] For each \(n\), the closure of the image of \(Z(G_n)\) in \(\mathcal{G}\) is open and coincides with \(C_{\mathcal{G}}(G_n)\).\N\item[(vii)] For each \(m\ge 1\) and each Hausdorff topological field \(F\), the only continuous homomorphism \(\mathcal{G} \rightarrow \mathrm{GL}_m (F)\) is the trivial one.\N\item[(viii)] Every finitely generated centerless subgroup of \(\mathcal{G}\) is discrete.\N\item[(ix)] The finitely generated centerless subgroups of \(\mathcal{G}\) fall into countably many abstract isomorphism classes.\N\end{itemize}