We study the convergence of volume-normalized Betti numbers in Benjamini–Schramm convergent sequences of nonpositively curved manifolds with finite volume. In particular, we show that if X is an irreducible symmetric space of noncompact type, $X\ne {\mathbb{H}}^3$, and $(M_n)$ is any Benjamini–Schramm convergent sequence of finite-volume X-manifolds, then the normalized Betti numbers $b_k(M_n)∕\mathrm{vol}(M_n)$ converge for all k.

As a corollary, if X has higher rank and $(M_n)$ is any sequence of distinct, finite-volume X-manifolds, then the normalized Betti numbers of $M_n$ converge to the $L^2$-Betti numbers of X. This extends our earlier work with Nikolov, Raimbault, and Samet, where we proved the same convergence result for uniformly thick sequences of compact X-manifolds. One of the novelties of the current work is that it applies to all quotients $M=\mathrm{\Gamma }\setminus X$ where Γ is arithmetic; in particular, it applies when Γ is isotropic.

Let ${\{S_i\}}_{i\in \mathrm{\Lambda }}$ be a finite contracting affine iterated function system (IFS) on ${\mathbb{R}}^d$. Let $(\mathrm{\Sigma },\mathit{\sigma })$ denote the two-sided full shift over the alphabet Λ, and let $\mathrm{\pi }:\mathrm{\Sigma }\to {\mathbb{R}}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic σ-invariant measure on Σ under π is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier–Young-type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.

We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.

We prove a general Russo–Seymour–Welsh (RSW) result valid for any invariant bond percolation measure on ${\mathbb{Z}}^2$ satisfying positive association. This means that the crossing probability of a long rectangle is related by a universal homeomorphism to the crossing probability of a short rectangle.