It is known that higher rank lattices have property (T) and also that lattices on 2-dimensional Euclidean buildings have property (T) provided the thickness $q+1$ of the building is large enough (which is a condition only in type $\tilde{C}_2$ and $\tilde{G}_2$). My question is about the best known bound that guarantees property (T).

Specifically Żuk states in his famous note that $q \ge 3$ is sufficient for type $\tilde{C}_2$ and that $q \ge 4$ is sufficient for type $\tilde{G}_2$. However, that statement is based on calculations performed by Garland and others (Garland's being the most readable) which only give sufficiency for $q \ge 7$ in type $\tilde{C}_2$ and for $q \ge 11$ in type $\tilde{G}_2$. Is there another justification for Żuk's claim? (Am I making a mistake?)

This is not purely hypothetical: Essert constructs lattices on $\tilde{C}_2$ buildings that lie in the region for which Żuk asserts property (T) but Garland's bounds do not guarantee it (as far as I can see).

Are there other references that give better bounds?

To turn all of this into a concrete question, let me ask this: Do all of the groups described by Essert (Section 0.2, Theorem 0.4) have property (T)?


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I don't have access to Zuk's note, but I remember finding an error in it when I read it (so this could be the same problem you found). He did improve on Garland in terms of thickness by taking average of the eigenvalues of the Laplacian of the links of two connected vertices - see the paragraph after the proof of Theorem 2.5 in Ballmann and Światkowski. However, that improvement does not grantee the results you quoted.

I managed to do a little better - see Table 2 (after Remark 4.1) in link, but that still does not give you $q=3$ for $\widetilde{C}_2$ lattices (only $q=4$).


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