[Paper Review] The mean Dehn function of abelian groups

Oleg Bogopolski, Enric Ventura|ArXiv.org|Jun 12, 2006

Geometric and Algebraic Topology4 references4 citations

TL;DR

This paper computes the mean Dehn functions for finitely generated abelian groups using elementary counting methods, showing that the three variations—$D_{osmean}(n)$, $D_{smean}(n)$, and $D_{mean}(n)$—are all bounded above by $Kn( ext{ln}\,n)^2$, where $K$ depends only on the group presentation and geodesic combing. This improves upon earlier subquadratic bounds and confirms Gromov's conjecture that mean Dehn functions grow significantly slower than standard Dehn functions.

ABSTRACT

While Dehn functions, D(n), of finitely presented groups are very well studied in the literature, mean Dehn functions are much less considered. M. Gromov introduced the notion of mean Dehn function of a group, $D_{mean}(n)$, suggesting that in many cases it should grow much more slowly than the Dehn function itself. Using only elementary counting methods, this paper presents some computations pointing into this direction. Particularizing them to the case of any finite presentation of a finitely generated abelian group (for which it is well known that $D(n)\sim n^2$ except in the 1-dimensional case), we show that the three variations $D_{osmean}(n)$, $D_{smean}(n)$ and $D_{mean}(n)$ all are bounded above by $Kn(\ln n)^2$, where the constant $K$ depends only on the presentation (and the geodesic combing) chosen. This improves an earlier bound given by Kukina and Roman'kov.

Motivation & Objective

• To investigate the growth rate of mean Dehn functions in finitely generated abelian groups, which are less studied than standard Dehn functions.
• To provide explicit upper bounds for three variations of the mean Dehn function: $D_{osmean}(n)$, $D_{smean}(n)$, and $D_{mean}(n)$.
• To confirm Gromov's conjecture that mean Dehn functions grow much more slowly than standard Dehn functions, particularly in abelian groups.
• To use elementary counting techniques and combinatorial estimates on word paths in Cayley graphs to derive asymptotic bounds.
• To improve upon the earlier subquadratic bound by Kukina and Roman’kov for the mean Dehn function of abelian groups.

Proposed method

• The authors define three variations of the mean Dehn function: $D_{osmean}(n)$, $D_{smean}(n)$, and $D_{mean}(n)$, based on averaging over words of length $n$ in the group.
• They use the Cayley graph of the group with respect to a finite generating set to model paths corresponding to words in the free monoid $A^*$.
• The analysis relies on counting paths without backtrackings (geodesics) in the Cayley graph of $\mathbb{Z}^r$, using known asymptotic formulas from Sharp (2001) for $\mathcal{N}_v'(n)$, the number of such paths ending at a given group element $v$.
• They apply Stirling’s approximation and asymptotic estimates for central binomial coefficients to analyze the number of closed paths of even length in $\mathbb{Z}^2$, which correspond to words representing the identity.
• The key technical step involves bounding the number of non-reduced words of length $n$ that represent the identity and have small area, using the structure of abelian presentations and the fact that $D(n) \sim n^2$ for abelian groups.
• The final bound is derived by combining these path-counting estimates with averaging arguments over the sphere $S_G(n)$ of words of length $n$ that evaluate to the identity in $G$.

Experimental results

Research questions

• RQ1 How do the mean Dehn functions $D_{osmean}(n)$, $D_{smean}(n)$, and $D_{mean}(n)$ grow for finitely generated abelian groups?
• RQ2 Can elementary counting methods yield subquadratic bounds for mean Dehn functions, as conjectured by Gromov?
• RQ3 What is the precise asymptotic behavior of the mean Dehn function in abelian groups, particularly in $\mathbb{Z}^r$?
• RQ4 How does the mean Dehn function compare to the standard Dehn function $D(n) \sim n^2$ in abelian groups?
• RQ5 Can the number of short-area words representing the identity be effectively bounded using path-counting techniques in the Cayley graph?

Key findings

• The mean Dehn functions $D_{osmean}(n)$, $D_{smean}(n)$, and $D_{mean}(n)$ for any finite presentation of a finitely generated abelian group are all bounded above by $Kn(\ln n)^2$, where $K$ depends only on the presentation and the chosen geodesic combing.
• This bound improves upon the earlier subquadratic bound established by Kukina and Roman’kov for the mean Dehn function of abelian groups.
• For the case of $\mathbb{Z}^2$, the number of closed paths of length $2n$ with no backtrackings is asymptotically $f_{2n} \sim \frac{4}{3(\sqrt{3}+1)\pi} \cdot \frac{3^{2n}}{2n}$, derived from Sharp’s local limit theorem.
• The asymptotic behavior of the number of geodesic paths in $\mathbb{Z}^r$ is used to estimate the number of words of length $n$ representing the identity and having small area, which is central to bounding the mean Dehn function.
• The authors confirm that the mean Dehn function grows strictly slower than the standard Dehn function $D(n) \sim n^2$, supporting Gromov’s conjecture on the slow growth of mean Dehn functions.
• The bound $O(n(\ln n)^2)$ is established through elementary counting and combinatorial estimates on path counts in the Cayley graph, without requiring advanced geometric or probabilistic tools.