Definition. Qualitative property (T) interface [connes-0003]

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HasKazhdanPropertyTHasRelativePropertyT

L∃∀NAGENTDRAFTED

Let \(G\) be a countable discrete group. A unitary representation \(\pi \) has almost-invariant unit vectors if every finite \(K\subset G\) and every \(\varepsilon >0\) admit a unit vector \(\xi \) satisfying \[\lVert \pi (g)\xi -\xi \rVert <\varepsilon \qquad (g\in K).\] The project defines property (T) by requiring every such representation to contain a nonzero invariant vector. Relative property (T) for a subgroup \(N\leq G\) requires every such representation to contain a nonzero vector fixed by \(N\).