Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

Bestvina, Mladen; Eskin, Alex; Wortman, Kevin

doi:10.4171/jems/419

Cited by 12 publications

(30 citation statements)

References 17 publications

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“…This looks a little different than the original statement of the theorem, but it is actually the same because we can immediately fill all of the bounded length relations at cost O(ℓ(w) 2 ).…”

Section: From Diagonal Blocks To Parabolicsmentioning

confidence: 83%

“…The most important progress in this direction is Young's Theorem 1.2, which proves that SL(p; Z) (a lattice in the rank p − 1 symmetric space SL(p; R)/SO(p)) has quadratic Dehn function for p at least 5. We must also mention a powerful result of Bestvina, Eskin, and Wortman [2,Corollary 5] which shows a polynomial Dehn function for groups such as SL(n; O K ) or Sp(2n; O K ) where O K is the ring of integers of a number field K which has at least 3 archimedean valuations (these groups are lattices in higher rank symmetric spaces. )…”

Section: Statement Of Main Theoremmentioning

confidence: 99%

“…At some point we will need to find a shortcut for e [1]− [2] This word exists because (writing S for {s} and T for {±t}), SpH S,T (Z) is cocompact in SpH S,T (R) (Lemma 7.3) and N S,T (R) is exponentially distorted in SpH S,T (R), as we can see by the following argument. Let v 1 , v 2 ∈ R T be the eigenbasis of T T , then for i = 1, 2, we know that u(z s ⊗ v i ) can be represented by a wordû(z s ⊗ v i ) of length O(log v i ).…”

Section: Defining Special Shortcuts For α a Short Rootmentioning

confidence: 99%

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A Dehn function for Sp(2n, ℤ)

Cohen

2017

J. Topol. Anal.

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Gromov conjectured that any irreducible lattice in a symmetric space of rank at least 3 should have at most polynomial Dehn function. We prove that the lattice Sp(2p; Z) has quadratic Dehn function when p ≥ 5. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted. Gromov's conjectureOur main theorem verifies a special case of a conjecture of Gromov about lattices in symmetric spaces. We now explain what this conjecture is, why it is widely believed, and how previous authors have attacked it. First though, we must recall some basic definitions about Dehn functions (for Riemannian manifolds), symmetric spaces, lattices, and horoballs.Dehn functions for Riemannian Manifolds. Let X be a complete, simply connected Riemmanian manifold. Then given a piecewise smooth loop γ : S 1 → X, we can find a piecewise smooth function f : D 2 → X whose restriction ∂f to the boundary is γ. The area of γ is defined to be the smallest area of any such filling, i.e., Area(γ) = inf{Area(f )|∂f = γ} 2 Main TheoremsWe will presently reduce our main theorem to Theorems 2.2, 2.1, and 2.3, which will be proved in other sections. To state these theorems, we first need to recall the definition of the standard maximal parabolic subgroups of the symplectic group along with some notions related to filling. Symplectic linear algebraHere we briefly recall the theory of symplectic linear algebra. (An inspired reference is [19]).Definition. The symplectic form ω is the bilinear form on R 2p given by ω(v, w) = v T J 0 w • e α (x)e α (y) = e α (x + y).

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Section: From Diagonal Blocks To Parabolicsmentioning

confidence: 83%

Section: Statement Of Main Theoremmentioning

confidence: 99%

Section: Defining Special Shortcuts For α a Short Rootmentioning

confidence: 99%

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A Dehn function for Sp(2n, ℤ)

Cohen

2017

J. Topol. Anal.

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“…Also, it has been shown that automatic groups have k-dimensional isoperimetric functions that are at most polynomial for every k, and that in the particular case when k = 1 their isoperimetric functions are always at most quadratic [ECH + ]. Recently there has been important progress on isoperimetric functions for lattices in higher rank semisimple groups, see [BEW,Leu2,You2].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial higher dimensional isoperimetry and divergence

Behrstock

Druţu²

2019

J. Topol. Anal.

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In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called "round" and "unfolded", provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer-Fleming inequality for finitely generated groups, the construction of examples of CAT (0)-groups with higher dimensional divergence equivalent to x d for every degree d [BD2], and a proof of the fact that for bi-combable groups the filling function above the quasi-flat rank is asymptotically linear [BD3].

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“…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.…”

mentioning

confidence: 99%

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

Leuzinger

Young

2016

Geom Dedicata

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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.

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Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

Cited by 12 publications

References 17 publications

A Dehn function for Sp(2n, ℤ)

A Dehn function for Sp(2n, ℤ)

Combinatorial higher dimensional isoperimetry and divergence

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

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