On the Čech cohomology of Morse boundaries

Abstract

We consider cusped hyperbolic $n-$manifolds, and compute Čech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced Čech cohomology with real coefficients vanishes in dimension at most $n-3$ and does not vanish in dimension $n-2$. A similar result holds for relatively hyperbolic groups with virtually nilpotent peripherals and Bowditch boundary homeomorphic to a sphere; these include all non-uniform lattices in rank$-1$ simple Lie groups.

Figures (8)

• Figure 1: A point in the boundary and its "projection" to the horosphere corresponding to the parabolic point $p$, as in the proof of item (1) of Lemma \ref{['lem:Psi']}.
• Figure 2: Illustration of the proof idea of item (2) of Lemma \ref{['lem:Psi']}.
• Figure 3: The points relevant for case (1) in the proof of Lemma \ref{['lem:shallow_barycentre']}.
• Figure 4: The decomposition of the discrete chain $d$ into parts close to parabolic points plus the rest $d_{\rm out}$.
• Figure 5: Illustrative diagram of the proof of part (\ref{['1']}) of Proposition \ref{['arbitrarily fine subdivisions 2']}, showing the inductive construction of the maps $\phi_i$.

Theorems & Definitions (82)

• Theorem A
• Conjecture 1
• Theorem B
• Definition 1.1
• Definition 1.2: Cones
• Lemma 1.3
• proof
• Definition 1.4: Face complex
• Definition 1.5
• Definition 1.6