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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, , and is any Benjamini–Schramm convergent sequence of finite-volume X-manifolds, then the normalized Betti numbers converge for all k.
As a corollary, if X has higher rank and is any sequence of distinct, finite-volume X-manifolds, then the normalized Betti numbers of converge to the -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 where Γ is arithmetic; in particular, it applies when Γ is isotropic.
Let be a finite contracting affine iterated function system (IFS) on . Let denote the two-sided full shift over the alphabet Λ, and let 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 a general Russo–Seymour–Welsh (RSW) result valid for any invariant bond percolation measure on 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.