Homology growth, hyperbolization, and fibering

TL;DR

The paper constructs closed odd-dimensional aspherical manifolds with word-hyperbolic fundamental groups that do not virtually fiber over a circle, using $\,\mathbb{F}_p$-homology growth as the obstruction. It develops a robust framework combining graph products, skew-field Betti numbers, and a hyperbolic reflection group trick (merging the Davis reflection group trick with Charney–Davis hyperbolization) to preserve hyperbolicity and residual finiteness while controlling homology growth via Mayer–Vietoris arguments. Key outputs include (i) existence results for 7-dimensional (and sometimes 5-dimensional) hyperbolic examples with nonvanishing $\,\mathbb{F}_p$-growth, (ii) explicit construction of a 7-manifold with special hyperbolic fundamental group whose $\underline{\beta}_4(\cdot;\mathbb{F}_p)>0$ for large $p$, and (iii) a barycentric-subdivision variant yielding refined control of $L^2$ and finite-field growth and connections to the $\,\mathbb{F}$-Singer property. Together, these results provide a concrete obstruction to virtual fibering in odd dimensions, advance the understanding of $L^2$ versus finite-field growth, and deliver a versatile toolbox for building hyperbolic manifolds with prescribed homological growth via virtual retracts and resolvent methods.

Abstract

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and -- for almost all primes $p$ -- $\mathbb{F}_p$-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic $7$-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.

Figures (4)

• Figure 1: The Davis chamber $K_L$ and its cubical structure. The cone vertex in $K_L$ corresponds to the empty simplex in $L$, and $\partial K_L$ consists of simplices/cubes not containing the cone vertex.
• Figure 2: The construction of $hP_{\partial}^N$. Starting with a triangulation of $\partial$, we first construct a locally CAT(0) cube complex $P_\partial$ and then a locally CAT(-1) space $hP_\partial$ where the vertices all have links isomorphic to $\partial N$. Then, each of these links is replaced with a copy of $N$ to form $hP_\partial^N$.
• Figure 3: An "unnaturally" embedded $hK_{{\mathop{\mathrm{Lk}}\nolimits(v)}}$ inside of $hK_{{\partial}}^{N}$.
• Figure 4: The space $hK_L^N$ and its $\pi_1(N)$-cover.

Theorems & Definitions (164)

• Theorem A
• Remark
• Theorem B
• Remark
• Theorem C
• Theorem D
• Theorem E
• Remark
• Theorem F
• Lemma 2.1