Group homology and ideal fundamental cycles

We prove that the group-homological version of the generalized Goncharov invariant of finite-volume locally rank one symmetric spaces determines their generalized Neumann-Yang invariant, which is defined using ideal fundamental cycles.

💡 Research Summary

The paper investigates two seemingly different invariants attached to finite‑volume locally rank‑one symmetric spaces: the group‑homological version of the generalized Goncharov invariant and the generalized Neumann‑Yang invariant defined via ideal fundamental cycles. The authors show that the former completely determines the latter, establishing a precise isomorphism between a homology class in the isometry group and a Borel‑Moore homology class represented by an ideal cycle.

The introduction outlines the difficulty of handling non‑compact locally symmetric spaces using ordinary singular chains, because the cuspidal ends contribute infinite volume. To overcome this, the authors adopt the notion of an ideal fundamental cycle, a chain that lives in the Borel‑Moore chain complex and incorporates the ideal points at infinity. This construction allows one to define a “geometric” volume class even for non‑compact manifolds.

In the background section the paper reviews Goncharov’s construction of higher‑order polylogarithmic invariants for fields, their interpretation as elements of the Bloch group, and the role of the Borel regulator in extracting real numbers (volumes) from these classes. It also recalls the Neumann‑Yang invariant, originally introduced for hyperbolic 3‑manifolds, which computes the volume of an ideal fundamental cycle by integrating the Bloch‑Wigner dilogarithm over the simplex parameters.

The main theorem states that for a finite‑volume locally rank‑one symmetric space (M) with orientation‑preserving isometry group (G), the natural map
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