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Filling functions of arithmetic groups. (English) Zbl 1493.20012

The Dehn function \(\delta _{G}\) of a finitely presented group \(G\) measures the complexity of the word problem in \(G\). More explicitly, \(\delta_{G}(\ell)\) is defined as the maximum number of of applications of relators necessary to reduce a word of length \(\ell\) that represents the identity to the trivial word.
The Dehn function can also be defined geometrically. Let \(X\) be a simply-connected manifold or a simplicial complex, \(\delta _{X}(\ell )\) is defined to be the maximum area necessary to fill a closed curve of length \(\ell\) by a disc.
There is a connection with the two definitions above. If the group \(G\) acts geometrically (cocompactly, properly discontinuously and by isometries) on the space \(X\), then the two Dehn functions \(\delta _{G}\) and \(\delta _{X}\) have the same asymptotic growth rate.
(See Section 2.4 for precise definitions.)
Filling volume functions generalize the Dehn function to higher dimensions.
Let \(X\) be an \((n-1)\)-connected metric space. The \(n\)-dimensional filling volume function \(FV_{X}^{n}\) measures the difficulty of filling \((n-1)\)-cycles in \(X\) by \(n\)-chains. It is harder to interpret this in terms of group theory, but it yields a quasi-isometry invariant. Let \(X\) and \(Y\) be quasi-isometric, highly connected and have bounded geometry (for instance, if they support a cocompact group action), then \(FV_{X}^{n}\) and \(FV_{Y}^{n}\) have the same asymptotic growth rate. Consequently, it gives rise to a group invariant: when a group \(G\) acts geometrically on \(X\), then let \(FV_{G}^{n}= FV_{X}^{n}\).
This definition of \(FV_{G}^{n}\) depends on the choice of \(X\), but its asymptotic growth rate is well defined.
In the case where the lattice \(\Gamma \subset G\) is uniform in \(G\), the filling functions are known [E. Leuzinger, Groups Geom. Dyn. 8, No. 2, 441–466 (2014; Zbl 1343.20046)] and, for non positively curved spaces, bounds for the functions \(\delta _{X}\) and \(FV_{X}^{n}\) are obtained [M. Gromov, J. Differ. Geom. 18, 1–147 (1983; Zbl 0515.53037); S. Wenger, Proc. Am. Math. Soc. 136, No. 8, 2937–2941 (2008; Zbl 1148.53031)].
If the lattice \(\Gamma\) is not uniform, such bounds are difficult to be obtained.
Thurston conjectured that the Dehn function of \(\Gamma=\mathrm{SL}_{k+1}(\mathbb{Z} )\) for \(k\geq 3 \) is quadratic. This has been proved for \(k\geq 4\) [R. Young, Ann. Math. (2) 177, No. 3, 969–1027 (2013; Zbl 1276.20053)]. M. Gromov extended this conjecture to higher-dimensional filling problems and arbitrary arithmetic lattices [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Proceedings of the symposium held at the Sussex University, Brighton, July 14–19, 1991. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)].
In this paper, the authors compute sharp bounds on the Dehn function and higher-dimensional filling volume functions of (irreducible) lattices in higher rank semisimple Lie groups and prove these conjectures.
Theorem 1.1.
Let \(\Gamma\) be an irreducible nonuniform lattice in a connected, center-free semisimple Lie group \(G\) without compact factors. Let \(k=\mathbb{R}-\operatorname{rank}(G)\), and suppose that \(k\geq 3\). Then the Dehn function of \(\Gamma\) is quadratic: \(\delta _{\Gamma}(L) \approx L^{2}\) for all \(L\geq 1\).
Theorem 1.2.
With \(\Gamma\) and \(k\) as in Theorem 1.1, we have \(FV_{\Gamma}^{n}(V) \approx V^{\frac{n}{n-1}}\) for all \(2\leq n<k\) and all \(V\geq 1\).
Theorem 1.2 holds only in dimensions below the rank of \(G\). It has been conjectured that the rank is a critical dimension for the isoperimetric behavior and that filling functions in the dimension of the rank grow exponentially [the first author and C. Pittet, Geom. Funct. Anal. 6, No. 3, 489–511 (1996; Zbl 0856.22013)].This has been shown in several cases (see [the first author and Pittet, loc. cit.; K. Wortman, Geom. Dedicata 151, 141–153 (2011; Zbl 1219.20027)]. Conversely, Gromov [loc. cit.] showed that filling functions of lattices in linear groups are at most exponential.
The next result of the authors confirms the conjecture in general.
Theorem 1.3.
With \(\Gamma\) and \(k\) as in Theorem 1.1, there exists \(c>0\) such that \(FV_{\Gamma}^{k}(V) \gtrsim \exp(cV^{\frac{1}{k-1}})\) for \(V\geq 1\).
A broader conjecture, based on the distortion of filling volumes, was proposed. K.-U. Bux and K. Wortman conjectured that \(S\)-arithmetic groups (defined over number fields or function fields) acting on products of symmetric spaces and Euclidean buildings are undistorted in dimensions below the geometric (or total) rank [Invent. Math. 167, No. 2, 355–378 (2007; Zbl 1126.20030)]. This is a strong generalization of a theorem of A. Lubotzky et al. on the distance distortion of lattices [Publ. Math., Inst. Hautes Étud. Sci. 91, 5–53 (2000; Zbl 0988.22007)]. Finiteness properties and filling invariants of \(S\)-arithmetic groups have been studied in several papers.
The authors confirm the Bux-Wortman conjecture in the case of nonuniform arithmetic groups defined over number fields (Theorem 1.4 in the paper).
Note that this theorem gives a new proof of the theorem of Lubotzky-Mozes-Raghunathan.
Most previous bounds for filling invariants of arithmetic lattices involve explicit constructions of cycles and chains in some thick part of \(X\). The bounds in this paper are based on a probabilistic method rather than on explicit constructions. Instead of constructing a single filling, it is shown that, when \(n<k\) and \(\alpha\) is an \((n-1)\)-cycle in \(X_{0}\) of mass \(V\), there is a large family of \(n\)-chains in \(X\) with boundary \(\alpha\) and mass at most \(V^{\frac{n}{n-1}}\). Then bounds are obtained by considering a random chain \(\beta\) drawn from this family.
The constructions in this paper follow the same broad outline as the constructions in [E. Leuzinger and R. Young, Geom. Dedicata 187, 69–87 (2017; Zbl 1366.22006)]. But, as the authors note, there are some crucial differences between the constructions here and those there. For details see in the paper.

MSC:

20F65 Geometric group theory
20P05 Probabilistic methods in group theory
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References:

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