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K-theory class of the first higher Kazhdan projection for amalgamated free products of finite groups

Determine whether, for any amalgamated free product of finite groups G = F *_K H, the class in K0(C(G)) of the first higher Kazhdan projection p1 (defined using the left regular representation and the combinatorial Laplacian Δ1) equals [p0^K] − [p0^F] − [p0^H], where p0^A denotes the averaging projection (1/|A|)∑_{a∈A} a viewed in C(G) via the natural inclusion of A into G.

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Background

The paper gives an explicit computation of the K0-class of the first higher Kazhdan projection for free products of finite cyclic groups G = Z_m * Z_n, namely [p1] = [1] − [p] − [q], where p and q are the averaging projections for Z_m and Z_n in C(G). For virtually free groups, it is established that p1 exists and its trace equals the first ℓ2-Betti number.

Motivated by this explicit result and the known formula for ℓ2-Betti numbers of amalgamated free products of finite groups, the authors ask whether an analogous expression in terms of averaging projections holds for general amalgamated free products G = F *_K H, with the proposed identity [p1] = [p0K] − [p0F] − [p0H] in K0(C(G)).

References

Let G be an amalgamated free product G = F \underset{K}{\ast} H of finite groups. We know by Lemma \ref{lem: existence} that p_1 exists and that $$\tau([p_1]) = \beta _1{2}(G) = \frac{1}{|K|}-\frac{1}{|F|}-\frac{1}{|H|}.$$ In view of Theorem \ref{thm: Z_n*Z_m} one may ask if [p_1] of G can be represented by

[p_1] = [p_0 ^ K] - [p_0 F] - [p_0H], where p_0's are the averaging projection associated with the involved finite groups?

— Higher Kazhdan projections and delocalised $\ell^ 2$-Betti numbers (2405.03837 - Pooya et al., 6 May 2024) in Section 4, Computations of higher Kazhdan projections and delocalised ℓ^2-Betti numbers, subsection “Free product of finite cyclic groups,” Question (near the end)