Quasi-isometric embeddings of Ramanujan complexes

Referee: [Abstract] Abstract and main theorem (presumably Theorem 1.1): the claim that the Ramanujan complexes for distinct primes do not quasi-isometrically embed into one another is false. Each complex is a finite simplicial complex with the path metric and is therefore a bounded metric space. For any two bounded metric spaces (X,d_X) and (Y,d_Y), the constant map f:X→Y with K=1 and C=max(diam(X),diam(Y)) satisfies (1/K)d_X(x,y)−C ≤ d(f(x),f(y)) ≤ K d_X(x,y)+C for all x,y, because the left-hand side is ≤0 while the right-hand side is ≥0. This directly falsifies the central claim and renders the cited rigidity theorems inapplicable to the finite quotients themselves.

Authors: We agree with the referee that the stated claim is false. The Ramanujan complexes arising in the Lubotzky-Samuels-Vishne construction are finite simplicial complexes equipped with the path metric, hence bounded metric spaces. As the referee notes, the constant map provides a quasi-isometric embedding between any two bounded spaces. This renders the main theorem, as written in the abstract and Theorem 1.1, incorrect, and the cited rigidity results cannot be applied directly to individual finite quotients in this manner. We will revise the abstract and the statement of the main theorem to assert instead that the box spaces associated to the sequences of Ramanujan complexes for distinct primes p and q do not quasi-isometrically embed into one another. This corrected formulation is consistent with the box-space rigidity theorem of Khukhro-Valette that we invoke, which applies to sequences of finite quotients rather than to any single finite complex. revision: yes

Referee: [Introduction] Introduction / proof outline: the manuscript assumes without explicit verification that the box-space rigidity of Khukhro-Valette and the building rigidity of Kleiner-Leeb/Fisher-Whyte transfer directly to the individual finite quotients arising in the Lubotzky-Samuels-Vishne construction. Because the rigidity statements concern either infinite buildings or box spaces built from sequences of quotients, this transfer step is load-bearing for the argument yet is not justified in detail.

Authors: We acknowledge that the manuscript does not supply a detailed justification for transferring the rigidity theorems to the individual finite quotients. The Euclidean building rigidity theorems of Kleiner-Leeb and Fisher-Whyte apply to the infinite buildings, while the box-space rigidity of Khukhro-Valette applies to sequences of quotients (box spaces). We will revise the introduction and the proof outline to explicitly describe the passage from the Lubotzky-Samuels-Vishne quotients to the associated box spaces, and to explain how non-embedding of the box spaces for distinct primes follows from Khukhro-Valette's theorem. We will add a short subsection clarifying the relevant limiting process and the precise objects to which each rigidity result is applied. revision: yes