Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

The arithmetic cocompact lattices constructed in (6.7.1) of Witte-Morris' book all have torsion-free finite index subgroups with arbitrarily large second Betti number.

I will briefly recall the construction because this is necessary for the answer. Let $F$ be a totally real number field with elements $a,b,t \in F$ such that $\sigma(a), \sigma(b), \sigma(t) < 0$ for all but one real $\sigma$ embedding of $F$. Let $L = F(\sqrt{t})$ be the degree $2$ extension with Galois group $\mathrm{Gal}(L/F) = \{\mathrm{id}, \tau\}$ and let $\mathcal{O}_L$ denote the ring of integers of $L$. Define $$h = \left(\begin{smallmatrix}a & & \\ & b & \\ & & -1\end{smallmatrix}\right)$$ The arithmetic group $$\Gamma = \{ g \in \mathrm{SL}_3(\mathcal{O}_L) \mid \tau(g)^T h g = h\} \subseteq \mathrm{SU}(h,L/F) $$ embeds as a cocompact lattice in $\mathrm{SL}_3(\mathbb{R})$ (via any one of the two embeddings $L \to \mathbb{R}$).

The non-trivial Galois automorphism $\tau$ of $L/F$ induces an automorphism of the algebraic group $\mathrm{SU}(h,L/F)$ which restricts to an automorphism $\tau$ of $\Gamma$. In particular, we get automorphisms $$\tau^j\colon H^j(\Gamma,\mathbb{C}) \to H^j(\Gamma, \mathbb{C})$$ in the cohomology. It is possible to calculate (or to bound) the Lefschetz number $$ L(\tau) = \sum_{j=0}^5 (-1)^j\mathrm{Tr}(\tau^j) $$ in the cohomology of $\Gamma$. The methods for this have been worked out by Jürgen Rohlfs (and others) in the 80's and 90's. The trick is to use the Lefschetz fixed point theorem on the associated locally symmetric space, which says that the Lefschetz number is the Euler characteristic of the set of fixed points. In the specific example the fixed point set should consist of a bunch of surfaces (and a couple of isolated points). Indeed, on $\mathrm{SL_3}(\mathbb{R})$ the automorphism is $g \mapsto h^{-1}(g^{-1})^Th$ and the group of fixed points is isomorphic to $SO(2,1)$. Since surfaces have non-zero Euler characteristic the method yields a lower bound for the cohomology. Here one can find a decreasing sequence of finite index subgroups $\Gamma_n \leq \Gamma$ such that $$ \sum_{j=0}^5 b_j(\Gamma_n) \gg [\Gamma:\Gamma_n]^{3/8} $$ as $n \to \infty$; see Theorem 4 in my article.

Asymptotically we have the same lower bound for $b_2(\Gamma_n)$. We have Poincare duality and by property (T) we know that $b_1(\Gamma_n) = 0 =b_4(\Gamma_n)$. Since $b_0(\Gamma_n) = b_5(\Gamma_n) = 1$ and we can only have interesting cohomology in degrees $2$ and $3$, and moreover $b_2(\Gamma_n) = b_3(\Gamma_n)$.

(In the case of non-cocompact lattices a similar argument has been carried out by Lee and Schwermer. I did not check whether the other examples of cocompact arithmetic lattices in $\mathrm{SL}_3(\mathbb{R})$ have a useful algebraic finite order automorphism.)

Disclaimer: I have not had time to check the details yet, but I think the following should work.

Let $f$ be the unique newform of level $30$, weight $2$ and trivial character. Let $\pi(f)$ be the automorphic representation of ${\rm GL}_2$ corresponding to $f$.

Let $\sigma$ be the Gelbart-Jacquet symmetric square lift of $\pi(f)$ to ${\rm GL}_3$. It has regular weight and is therefore cohomological.

Let $D$ be the division algebra over $\mathbb{Q}$ of dimension $9$ with invariants $1/3$ at $2$, $2/3$ at $3$ and $0$ elsewhere. Let $\Gamma$ be the group of elements of reduced norm $1$ in a maximal order of $D$. This is a cocompact lattice in ${\rm SL}_3(\mathbb{R})$. Let $\tau$ be the Jacquet-Langlands transfer of $\sigma$ to $D^\times$, which exists because $\sigma$ is Steinberg at $2$ and $3$ (by work of Badulescu), and is still cohomological.

Then by Matsushima's formula, $\tau$ contributes nontrivially to the $H^2$ and $H^3$ of some torsion-free congruence subgroup of $\Gamma$.