Dehn functions of groups with filiform subgroups

This thesis provides new insights on the asymptotic geometry of locally compact groups with a particular focus on their Dehn functions.

In this work, we focus on the study of the Dehn functions of compact and finitely generated groups that contain so-called filiform groups as subgroups. Filiform groups play a central role in understanding the asymptotic geometry of the groups we investigate. We present several classes of groups where the Dehn functions are determined by the Dehn function of a maximal filiform subgroup. This leads to the question of whether this is a general principle. We answer this question negatively by identifying an interesting phenomenon in the context of nilpotent groups. Specifically, we observe that in certain central products of nilpotent groups, where filiform groups are used as building blocks, their Dehn functions are smaller than those of the maximal filiform subgroup.

The main results of this work consist of determining the precise Dehn functions of a large class of nilpotent groups and of natural families of mapping tori of right-angled Artin groups, RAAGs for short. The study of Dehn functions of nilpotent groups can be understood in the context of the conjectural quasi-isometry classification of nilpotent groups (see Conjecture 1); whereas the study of the Dehn functions of mapping tori of RAAGs is motivated by a question of Vogtmann [PR19, p.2] motivated by the work in [BP94, BG96, BG10].
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In the first part of this work, we establish upper bounds on the Dehn functions of an uncountable family of nilpotent groups arising as central products of nilpotent groups (see Theorems A and B). These results and the methods used generalise those appearing in [LIPT23]. Moreover, these results provide further supporting evidence for a conjecture made by the authors in [LIPT23, Conjecture 11.13] (see Conjecture 3). The results presented here combined with the lower bounds on Dehn functions obtained in [GMLIP23] (see Section 3.6 for an overview) provide us with the precise Dehn functions of these central products. This has the consequence of significantly extending the class of pairs of groups with bilipschitz equivalent asymptotic cones and different Dehn functions (see Corollary C). Since the Dehn function is a quasi-isometry invariant we distinguish such groups from each other up to quasi-isometry. An important feature is that it is possible to quantify the failure of the existence of a quasi-isometry in n Corollary C by means of a computable algebraic invariant defined by Cornulier [Cor17]. As consequence of Theorems A and B we show that this invariant is asymptotically optimal for a particular class of simply connected nilpotent Lie groups (see Theorem D).

In the second part of this work we determine Dehn functions of natural families of mapping tori of RAAGs, namely semidirect products of a right-angled Artin group $G$ by $\mathbb{Z}$. We list our main results obtained in this part:

1. A novel result in this work is the full characterisation of the Dehn functions of mapping tori of the RAAG $\mathbb{Z}^2$ $\ast$ $\mathbb{Z}^2$ (see Theorem E). To the best of our knowledge, this provides a full description of the Dehn functions for an entire class of mapping tori of RAAGs, for which their Dehn functions have not been studied or known before. Moreover, the methods we use to obtain these Dehn functions, modulo a fairly reasonable Conjecture 5, which holds for $\mathbb{Z}^2$ $\ast$ $\mathbb{Z}^2$ (see Lemma 4.3), have the advantage to be generalisable to obtain the Dehn functions of mapping tori of $\mathbb{Z}^m$ $\ast$ $\mathbb{Z}^n$ with $m, n \geq 2$.
2. We obtain a full characterisation of the Dehn functions of mapping tori of the direct product of finitely many finite rank free groups: $F_{m_1} \times \ldots \times F_{m_k}$ with $k \geqslant 2$ (see Theorem F and Theorem 4.7). This generalises the result in [PR19, Pue16, Thereom A] for the case $k=2$.
3. Finally, we present an alternative new proof to the one in [PR19, Section 6] for the mapping torus of $\mathbb{Z}^2$ $\ast$ $\mathbb{Z}$ with quadratic Dehn function (see Section 4.4 and Theorem 4.10). We anticipate that the strategy used to prove this case can serve as a general approach to fully characterise the Dehn functions of the mapping tori of right-angled Artin groups $\mathbb{Z}^m$ $\ast$ $\mathbb{Z}$ with $m \geqslant 2$.

We would like to mention that together the results and methods from 1. and 3., mentioned above, outline a promising path to obtaining a full characterisation of the Dehn functions of the mapping tori of the RAAGs $\mathbb{Z}^m$ $\ast$ $\mathbb{Z}^n$ with $m, n \geqslant 1$.