On regular subgroups of $\mathsf{SL}_{3}(\mathbb{R})$

Our discussion is motivated by the following question of M. Kapovich, also considered by D. Long and A. Reid.

We remark that $\mathsf{SO}_{3,1}(\mathbb{Z})$ contains copies of $\mathbb{Z}^{2}*\mathbb{Z}$, and hence so does $\mathsf{SL}_{n}(\mathbb{Z})$ for each $n\geqslant 4$. While we do not resolve Question 1.1, we establish the following.

Regularity (defined with respect to a parabolic subgroup) is a form of discreteness for subgroups of, or representations into, noncompact semisimple Lie groups that coincides with discreteness in the rank-one setting, but is strictly stronger in higher rank. These subgroups already appear in work of Benoist [1], and are the divergent subgroups of Guichard and Wienhard [7]; see Section 2 for the precise definition. For instance, the aforementioned copies of $\mathbb{Z}^{2}*\mathbb{Z}$ in $\mathsf{SL}_{n}(\mathbb{Z})$ for $n\geqslant 4$ are regular, as are Anosov representations of Gromov-hyperbolic groups [12, 7] and, more generally, relative Anosov representations of relatively hyperbolic groups [9, 17]. On the other hand, a lattice in a Cartan subgroup of $\mathsf{SL}_{3}(\mathbb{R})$ is not regular. Indeed, we show that regular $\mathbb{Z}^{2}$ subgroups of $\mathsf{SL}_{3}(\mathbb{R})$ are of a very particular form. Recall that a (resp., minimal, maximal) horospherical subgroup of $\mathsf{SL}_{3}(\mathbb{R})$ is by definition the unipotent radical of a (resp., maximal, minimal) proper parabolic subgroup of the latter.

It follows from Theorem 1.3 and results¹¹1In greater detail, suppose that $\Gamma<\mathsf{SL}_{3}(\mathbb{R})$ is discrete and Zariski-dense, and that some minimal horospherical subgroup $U$ of $\mathsf{SL}_{3}(\mathbb{R})$ is $\Gamma$-compact, where, following Oh [14], we say that a closed subgroup $H$ of $\mathsf{SL}_{3}(\mathbb{R})$ is $\Gamma$-compact if $H/(H\cap\Gamma)$ is compact. Then Oh exhibits in [14, Prop. 4.1] a ($\Gamma$-compact) maximal horospherical subgroup $V$ of $\mathsf{SL}_{3}(\mathbb{R})$ containing $U$ such that the other minimal horospherical subgroup $U^{\prime}$ of $\mathsf{SL}_{3}(\mathbb{R})$ contained in $V$ is also $\Gamma$-compact. By Zariski-density of $\Gamma$, there is some $\gamma\in\Gamma$ such that the $\Gamma$-compact minimal horospherical subgroups $U$ and $\gamma U^{\prime}\gamma^{-1}$ are opposite to one another [14, Prop. 3.6]. One now applies the main theorem of [13] to conclude that $\Gamma$ is commensurable to a conjugate of $\mathsf{SL}_{3}(\mathbb{Z})$. See also Benoist’s survey [2, Prop. 4.1]. of Oh [13, 14] that any Zariski-dense discrete subgroup of $\mathsf{SL}_{3}(\mathbb{R})$ containing a regular $\mathbb{Z}^{2}$ is in fact commensurable to an $\mathsf{SL}_{3}(\mathbb{R})$-conjugate of $\mathsf{SL}_{3}(\mathbb{Z})$. Theorem 1.2 now follows since any discrete $\mathbb{Z}^{2}\ast\mathbb{Z}$ in $\mathsf{SL}_{3}(\mathbb{R})$ is necessarily Zariski-dense, while $\mathbb{Z}^{2}\ast\mathbb{Z}$ cannot be realized as a lattice in $\mathsf{SL}_{3}(\mathbb{R})$, for instance, because groups of the latter form enjoy Kazhdan’s property (T) [11] (see also Furstenberg [6]). Moreover, as regularity is inherited by subgroups, and since $\mathsf{SL}_{3}(\mathbb{Z})$ contains a lattice in a Cartan subgroup of $\mathsf{SL}_{3}(\mathbb{R})$, we deduce the following.

We remark that if $F$ is a lattice in a minimal horospherical subgroup of $\mathsf{SL}_{3}(\mathbb{R})$, then the limit set of $F$ in the Furstenberg boundary of $\mathsf{SL}_{3}(\mathbb{R})$ is the set of all projective flags of the form $(z,\ell)$, where either the point $z\in\mathbb{P}(\mathbb{R}^{3})$ is fixed and $\ell\subset\mathbb{P}(\mathbb{R}^{3})$ varies among all projective lines in $\mathbb{P}(\mathbb{R}^{3})$ containing $z$, or the projective line $\ell$ is fixed and $z$ varies among all points of $\ell$; for the precise notion of limit set used here, see Section 2. Thus, another consequence of Theorem 1.3 is that a relative Anosov subgroup $\Gamma$ of $\mathsf{SL}_{3}(\mathbb{R})$ contains no $\mathbb{Z}^{2}$ subgroups, since the limit set of such $\Gamma$ in the Furstenberg boundary of $\mathsf{SL}_{3}(\mathbb{R})$ is antipodal in the language of Kapovich–Leeb–Porti; see [9, 17].

It is known that a group admitting a relative Anosov representation is relatively hyperbolic with respect to a family of virtually nilpotent subgroups; see [9, 17]. Since polycyclic groups that lack $\mathbb{Z}^{2}$ subgroups are virtually cyclic, and since groups that are hyperbolic relative to virtually cyclic (more generally, hyperbolic) subgroups are themselves hyperbolic, we conclude the following from the previous paragraph.

In fact, in forthcoming work [16] of the second-named author with F. Zhu, Corollary 1.5 is used to prove the stronger statement that a relative Anosov subgroup of $\mathsf{SL}_{3}(\mathbb{R})$ is virtually a free group or a surface group.

The relevance of Theorem 1.3 to Question 1.1 is further explained by the following proposition.

Thus, if there had been a regular $\mathbb{Z}^{2}$ in $\mathsf{SL}_{3}(\mathbb{Z})$ with “small” limit set in $\mathbb{P}(\mathbb{R}^{3})$—a scenario that is ruled out by Theorem 1.3—then Proposition 1.6 would have furnished a $\mathbb{Z}^{2}*\mathbb{Z}$ subgroup of $\mathsf{SL}_{3}(\mathbb{Z})$, and even a regular such subgroup by work of Dey and Kapovich [4, Thm. 3.2].

In light of Corollary 1.4, a result [3, Thm. 1.1] of the second-named author with R. Canary asserting that Anosov subgroups of $\mathsf{SL}_{3}(\mathbb{R})$ are virtually isomorphic to Fuchsian groups, and aforementioned forthcoming work of the second-named author with Zhu, the following question seems natural.

For two sequences $(a_{k})_{k\in\mathbb{N}}$ and $(b_{k})_{k\in\mathbb{N}}$ of positive real numbers, we write $a_{k}\asymp b_{k}$ (resp., $a_{k}=O(b_{k})$) if there is a constant $C>1$ such that $C^{-1}a_{k}\leqslant b_{k}\leqslant Ca_{k}$ (resp., $a_{k}\leqslant Cb_{k}$) for every $k$.

Throughout this section, let $G$ be a finite-center real semisimple Lie group with finitely many connected components and maximal compact subgroup $K<G$, and let $X=G/K$ be the associated symmetric space. Let $P$ be a proper parabolic subgroup of $G$, so that $P$ is the stabilizer in $G$ of a point $z\in\partial_{\infty}X$, where $\partial_{\infty}X$ denotes the visual boundary of $X$. Pick a point $o\in X$, and let $\xi$ be the geodesic ray in $X$ emanating from $o$ in the class of $z$. Fix also a Weyl chamber $\overline{\mathfrak{a}}^{+}\subset X$ for $G$ in $X$ with origin $o$ containing the ray $\xi$. A sequence $(g_{n})_{n\in\mathbb{N}}$ in $G$ is $P$-regular if the vector-valued distances $d_{\overline{\mathfrak{a}}^{+}}(o,g_{n}o)$ diverge from each wall of $\overline{\mathfrak{a}}^{+}$ not containing $\xi$. This notion is independent of all the choices made after specifying the parabolic subgroup $P$. If $\Gamma$ is a discrete group, a representation $\rho:\Gamma\rightarrow G$ is called $P$-regular if for every sequence $(\gamma_{n})_{n\in\mathbb{N}}$ in $\Gamma$ with $\gamma_{n}\rightarrow\infty$, the sequence $(\rho(\gamma_{n}))_{n\in\mathbb{N}}$ is $P$-regular. We remark that such a representation has finite kernel and discrete image, and is moreover $P^{\mathrm{opp}}$-regular, where $P^{\mathrm{opp}}$ denotes a parabolic subgroup opposite to $P$. A subgroup $\Delta$ of $G$ is called P-regular if the inclusion $\Delta\xhookrightarrow{}G$ is $P$-regular. Notice that if a subgroup $\Gamma$ of $G$ is $P$-regular then so are all subgroups of $\Gamma$. For more background, see Kapovich–Leeb–Porti [10], as well as earlier work of Guichard–Wienhard [7] where the notion of $P$-regularity appears instead as $P$-divergence.

We will also use the following characterization of $P$-regularity in terms of the dynamics on the flag manifold $G/P$. A sequence $(g_{n})_{n\in\mathbb{N}}$ is called $P$-contracting if there are points $z^{+}\in G/P$ and $z^{-}\in G/P^{\mathrm{opp}}$ such that $g_{n}$ converges uniformly on compact subsets of $C(z^{-})$ to the constant function $z^{+}$, where $C(z^{-})$ denotes the set of all points in $G/P$ opposite to $z^{-}$. In this case, we write $(g_{n})_{n}^{+}:=z^{+}$.

The limit set of a subgroup $\Gamma<G$ in the flag manifold $G/P$, denoted by $\Lambda_{\Gamma}^{P}$, is by definition the set of $(\gamma_{n})_{n}^{+}\in G/P$ for all $P$-contracting sequences $(\gamma_{n})_{n\in\mathbb{N}}$ in $\Gamma$. If $P$ is conjugate to $P^{\mathrm{opp}}$, two subsets $\Lambda_{1},\Lambda_{2}\subset G/P$ are antipodal if each element of $\Lambda_{1}$ is opposite to each element of $\Lambda_{2}$.

The proof of the following lemma uses the fact that, for a matrix $g=(g_{ij})_{i,j=1}^{3}$ in $\mathsf{SL}_{3}(\mathbb{C})$, one has $\frac{1}{\sqrt{3}}||g||_{2}\leqslant\sigma_{1}(g)\leqslant||g||_{2}$, where $||g||_{2}:=(\sum_{i,j=1}^{3}|g_{ij}|^{2})^{1/2}$ is the $\ell^{2}$-matrix norm of $g$.

To prove Proposition 1.6, we use the following variant of the ping-pong lemma. Similar arguments appear in work of Dey and Kapovich [4], but we include them here for the convenience of the reader.