Cohomology of Fuchsian Groups and Non-Euclidean Crystallographic Groups

TL;DR

This work computes the cohomology of geometrically finite 2-dimensional NEC groups, with a complete ring structure in the special case of cocompact Fuchsian groups. It deploys a Cartan–Leray type spectral sequence on a contractible model $\mathbb{R}\mathbf{H}^2\cup\Lambda(\Gamma)$ and leverages stabilizer data from NEC signatures to produce explicit $H^q(\Gamma)$ in all cases, distinguishing orientable vs non-orientable quotients and presence of cusps/boundaries. The results include precise free and torsion components, parity conditions modulo 4, and explicit ring decompositions using torision components $R_q$ and invariant factors $t_j$, $w_j$, and $q_k$. These cohomological descriptions illuminate the relationship between group structure, orbifold topology, and cohomological invariants, and confirm compatibility with $L^2$-Betti number results while showing that cohomology does not determine the underlying Fuchsian groups.

Abstract

For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.

Figures (5)

• Figure 1: In (a) we have a fundamental domain for an NEC group of signature $(1,0,+,[],\{(m,n),()\})$. The topological quotient of $\mathbb{R}\textbf{H}^2$ is homeomorphic to a torus with two open discs removed. In the orbifold structure of the quotient we have two cone points on one of the two boundary components. In (b) we have a fundamental domain for a Fuchsian triangle group of signature $(0,0,+,[p,q,r],\{\})=[0,0;p,q,r]$ for $p^{-1}+q^{-1}+r^{-1}<1$. The topological quotient is homeomorphic to a sphere. In the orbifold structure we have three cone points. In (c) we have a fundamental domain of a Fuchsian NEC group of signature $(0,1,+,[m,n],\{\})=[0,1;m,n]$ for $m+n>4$. The topological quotient is homeomorphic to a punctured sphere. In the orbifold structure we have two cone points in the interior of the punctured sphere. In (d) we have a fundamental domain for a non-orientable NEC group of signature $(4,0,-,[m,n],\{\})$ for $m,n\geq2$. The topological quotient is homeomorphic to a non-orientable surface. In the orbifold structure we have two cone points.
• Figure 2: The $E^1$-page of the spectral sequence for a Fuchsian group.
• Figure 3: The $E^2$-page of the spectral sequence for a Fuchsian group.
• Figure 4: The $E^1$-page of the spectral sequence for an orientable NEC group with cusps and boundary.
• Figure 5: The $E^2$-page of the cohomological spectral sequence for a cocompact Fuchsian group. Here the element $x_j$ is additive torsion of order $m_j$.

Theorems & Definitions (15)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 3.1
• proof : of Theorem \ref{['theorem.main']}\ref{['theorem.main.1']}
• Corollary 4.1
• proof : of Theorem \ref{['theorem.main']}\ref{['theorem.main.4']}
• Corollary 4.2
• Theorem 4.3