Non-LERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups (Q2631523)

A subgroup $H$ of a group $G$ is \textsl{separable} if for every $g\in G\setminus H$, there is a finite index subgroup $K$ of $G$ containing $H$ but not $g$. A group is \textsl{LERF (locally extended residually finite)} if all its finitely generated subgroups are separable. The notion has particular interest in the case when $G$ is the fundamenental group of a manifold, for it can be used to promote essential immersions to embeddings in some finite cover of the manifold. Surface groups are known to be LERF by work of \textit{P. Scott} [J. Lond. Math. Soc., II. Ser. 17, 555--565 (1978; Zbl 0412.57006)] as are fundamental groups of $3$-manifolds admitting either a Seifert fibred structure [loc. cit.] or a hyperbolic one [\textit{I. Agol}, Doc. Math. 18, 1045--1087 (2013; Zbl 1286.57019), \textit{D. T. Wise}, From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry. Providence, RI: American Mathematical Society (AMS); Washington, DC: Conference Board of the Mathematical Sciences (CBMS) (2012; Zbl 1278.20055)]. \par The author provides examples of non LERF groups suggesting that the LERF property is a peculiarity of fundamental groups of geometric manifolds of dimension at most $3$. \par A first result in the paper shows that fundamental groups of irreducible toroidal $3$-manifolds containing a hyperbolic piece in their JSJ decomposition are not LERF; moreover the author provides an explicit geometric description of a non separable subgroup (isomorphic to a free group or a surface group). This generalises work of \textit{Y. Liu} [Trans. Am. Math. Soc. 369, No. 2, 1237--1264 (2017; Zbl 1352.57001)] on mapping tori of reducible maps. \par Together with a previous result of \textit{G. A. Niblo} and \textit{D. T. Wise} [Proc. Am. Math. Soc. 129, No. 3, 685--693 (2001; Zbl 0967.20018)] on graph manifolds, the above imples that the fundamental group of an irreducible $3$-manifold is LERF if and only if the manifold is geometric, that is, admits a geometric structure modelled on one of Thurson's eight $3$-dimensional geometries. \par Using the fact that the LERF property is preserved when taking subgroups, by passing to finite covers and appropriately chosen submanifolds the author is able to reduce the proof of the aforementioned result to the case of manifolds admitting a specific JSJ decomposition (two geometric pieces, at least one of them hyperbolic, and with dual graph consisting of two vertices connected by two edges). The techniques used in the proof also provide ways to construct non LERF groups starting from hyperbolic $3$-manifold groups, notably by identifying a simple closed geodesic of a first manifold to a simple closed geodesic in a second one, or by taking HNN-extensions of such groups over infinite cyclic subgroups. \par Finally, exploiting the presence of several totally geodesic submanifolds inside arithmetic manifolds, together with generalisations of the arguments just described, the author shows that the fundamental groups of non compact arithmetic hyperbolic manifolds of dimension at least $4$, as well as of compact arithmetic hyperbolic manifolds of dimension at least $5$ (with the exception of the $7$-dimensional ones defined by octonions) are not LERF. Some other examples of non-arithmetic hyperbolic manifold groups that are not LERF are also provided.