A subgroup $H$ of a group $G$ is *malnormal* if $gHg^{-1}\cap H=\{e\}$ for all $g\in G$ with $g\notin H$. It is *almost malnormal* if we merely require $gHg^{-1}\cap H$ to be finite.


I am wondering whether $\mathrm{SL}(n,\mathbb{Z})$, or $\mathrm{PSL}(n,\mathbb{Z})$, for $n\geq 2$, have (almost) malnormal non-abelian free subgroups. And why or why not?


More generally, I'd be happy to know a bit more about when countably infinite groups contain malnormal non-abelian free subgroups. Thanks!


Edit: As pointed out in Moishe's answer, this really should be a question for $\mathrm{PSL}(n,\mathbb{Z})$, not $\mathrm{SL}(n,\mathbb{Z})$, to avoid the possibility of (trivially) conjugating with $-I$.



Does $\mathrm{SL}(n,\mathbb{Z})$ have an (almost) malnormal free subgroup?
First of all, you have to work with $G=PSL(2, {\mathbb Z})$ and not $SL(2, {\mathbb Z})$, for otherwise the claim is clearly false. Then $G$ is a nonelementary hyperbolic group with trivial maximal finite normal subgroup. According to Lemma 8 in


*Minasyan, Ashot; Olshanskii, Alexander Yu.; Sonkin, Dmitriy*, [**Periodic quotients of hyperbolic and large groups.**](https://arxiv.org/abs/0804.3328), Groups Geom. Dyn. 3, No. 3, 423-452 (2009). [ZBL1234.20051](https://zbmath.org/?q=an:1234.20051).


the group $G$ contains a malnormal subgroup which is also free of rank 2.


As for the group $PSL(n, {\mathbb Z})$, $n\ge 3$, the situation is unclear. I do know that it is impossible to find an almost malnormal Zariski dense subgroup because such subgroups always contain real-regular elements [2] and the latter have centralizers which are not virtually cyclic [1].


The references are


[1] *Prasad, Gopal; Raghunathan, M. S.*, [**Cartan subgroups and lattices in semi-simple groups**](https://doi.org/10.2307/1970790), Ann. Math. (2) 96, 296-317 (1972). [ZBL0245.22013](https://zbmath.org/?q=an:0245.22013).


[2] *Prasad, Gobal*, [**$\mathbb{R}$-regular elements in Zariski-dense subgroups**](https://doi.org/10.1093/qmath/45.4.542), Q. J. Math., Oxf. II. Ser. 45, No. 180, 541-545 (1994). [ZBL0828.22010](https://zbmath.org/?q=an:0828.22010).


