Reduced 1-cohomology and relative property (T)

TL;DR

This paper explores the relationship between reduced 1-cohomology and relative property (T), showing that the vanishing of cohomology restriction maps does not necessarily imply relative property (T).

Contribution

It provides a new characterization of group pairs with vanishing cohomology restriction maps, highlighting their distinction from relative property (T).

Findings

Vanishing of cohomology restriction maps is not equivalent to relative property (T).

Characterization of group pairs with specific cohomological properties.

Clarification of the relationship between cohomology and property (T).

Abstract

Shalom characterized property (T) in terms of the vanishing of all reduced first cohomology. We characterize group pairs having the property that the restriction map on all first reduced cohomology vanishes. We show that, in a strong sense, this is inequivalent to relative property (T).