The distortion dimension of $$\mathbb Q$$-rank 1 lattices

Abstract

Let X = G/K be a symmetric space of noncompact type and rank k ≥ 2. We prove that horospheres in X are Lipschitz (k − 2)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible Q-rank-1 lattice Γ in a linear, semisimple Lie group G of R-rank k is k − 1. That is, given m < k − 1, a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) Γ, and a (m + 1)-ball B in X (or G) filling S , there is a (m + 1)-ball B in Γ filling S such that vol B ∼ vol B. In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension k − 1.Consider now a group G of the form G = m i=1 G i (k i ), where the k i are locally compact, non-discrete fields and the G i are connected, absolutely almost simple algebraic groups defined over k i . Let Γ be an irreducible lattice in G. In a remarkable paper [15], Lubotzky, Mozes and Raghunathan proved that if the total rank k ofIn [4], Bux and Wortman conjectured a far reaching generalization of this result. In order to formulate it, let X be the product of irreducible symmetric spaces and Euclidean buildings on which Γ acts. The total rank of G is then equal to the maximal dimension of an isometrically embedded Euclidean space in X, which we call the geometric rank of X and denote by geo-rank(X). For some point x ∈ X and a real number r define the the following thickening of the orbit Γ · x in X X(r) := {y ∈ X | d(y, Γ · x) ≤ r}.Note that by the Milnor-Švarc lemma, the induced inner metric on X(r) is quasi-isometric to (Γ, d Γ ). Following [4] we define Γ as being undistorted up to dimension m if: given any r ≥ 0, there exist real numbers r ≥ r, λ ≥ 1, and C ≥ 0 such that for any k < m and any Lipschitz k-sphere S ⊂ X(r), there is a Lipschitz (k + 1)-ball B Γ ⊂ X(r ) with ∂B Γ = S and volume(B Γ ) ≤ λ volume(B X ) + C for all Lipschitz (k + 1)-balls B X ⊂ X with ∂B X = S . The distortion dimension of Γ is then defined as dis-dim(Γ) = max{m | Γ is undistorted up to dimension m}.The conjecture of Bux and Wortman posits that dis-dim(Γ) = geo-rank(X) − 1. See [5], [20] for recent progress on this conjecture. The chief goal of the present paper is to prove the Bux-Wortman conjecture for Q-rank 1 arithmetic groups in linear, semisimple groups defined over number fields, i.e. finite extensions of Q. For such lattices the space X above is a symmetric space of noncompact type; that is, there are no building factors. In our proof, it will be convenient to replace the subset X(r) by the complement of a countable union of horoballs in X (see [13], Thm. 3.6). Like X(r), this is quasi-isometric to (Γ, d Γ ). A crucial fact is that for Q-rank 1 lattices these horospheres are disjoint.Theorem A (Distortion dimension). The distortion dimension of an irreducible Q-rank 1 lattice in a linear, semisimple Lie group of R-rank k is k − 1. If k ≥ 2, then such an arithmetic 1. E(∆) contains a copy ∆ of ∆ at its center.

Filling functions of arithmetic groups

“…Similar constructions were used in [LP96] and [Wor11] to find exponential lower bounds in other groups. Upper bounds on filling invariants of arithmetic lattices and solvable groups have been found in [Dru04,You13,Coh17,LY17,BEW13], and [CT17]. These bounds typically combine explicit constructions of chains that fill cycles of a particular form with ways to decompose arbitrary cycles into pieces of that form.…”

“…Sketch of proof. The constructions in this paper follow the same broad outline as the constructions in [LY17]. As in that paper, we build fillings of cycles by gluing together large simplices.…”

“…There are two main differences between the constructions in this paper and those in [LY17]. First, instead of constructing a single map Ω, we construct a family of maps Ω D , parametrized by a certain tuple D of chambers in the geodesic boundary X ∞ , then construct a random variable R which takes values in the set of such tuples.…”

Undistorted fillings in subsets of metric spaces

“…Moreover, certain horospheres in symmetric spaces of noncompact type of rank k or in the product of k proper CAT(0) spaces have (LC k−2 ). This has been established in [28] and [30], respectively. The jet space Carnot groups (J s (R k ), d c ) have (LC k−1 ); see [42].…”

“…These results are consequences of a more general theorem about isoperimetric subspace distortion. Isoperimetric distortion of subspaces was briefly addressed by Gromov in [18] and has recently been studied in the articles [10], [45], [28], [29] in connection with conjectures of Thurston, Gromov, and Bux-Wortman. Roughly speaking, a subspace X of a metric space Y is (isoperimetrically) undistorted up to dimension k + 1 if m-cycles in X with 0 ≤ m ≤ k can be filled almost as efficiently by (m + 1)-chains in the subspace X as they can be filled in the ambient space Y.…”