Geometric presentations of Lie groups and their Dehn functions (Q2400806)

This is a long and important contribution, which addresses the problem of the determination of the Dehn function for connected Lie groups and, more generally, it describes the behaviour of this function for compactly presented groups. For a simply connected Lie group $G$ possessing a left-invariant Riemannian metric, the area of a loop $\gamma$ is the infimum of areas of filling discs, and the Dehn function of $G$ can be defined as the supremum of areas of loops of length at most $r$. The knowledge of the Dehn function and their growth are important for various aspects, which are related to some fundamental contributions of M. Gromov in geometric group theory. Since compact abelian Lie groups are isomorphic to $\mathbb{T}^n \oplus E$, where $\mathbb{T}^n$ denotes the direct sum of $n$ tori $\mathbb{T}$ and $E$ is finite abelian, one could expect that the Dehn function has a very slow rate of growth, but the moment we remove the assumption of being compact abelian, this intuition is no longer justified and a series of observations and technical results must be discussed. The first main result of the present contribution shows that a connected Lie group $G$ has Dehn function either exponential or polynomially bounded (see Theorem A). The arguments of the proofs involve the techniques of the combinatorial group theory and of the Riemannian geometry, in fact they allow to prove much more. For instance, Theorem C shows that an algebraic group over a $p$-adic field has Dehn function at most cubic, otherwise $G$ is not compactly presented. The notion of ``compact presentations'' are due to H. Abels and extend the notion of profinite presentations, which are well known in the context of profinite groups. Joining the notion of Dehn function to compact presentation is probably the central idea of this contribution and its formalization requires several conceptual steps. \par Theorems D and F deal with the behaviour of Dehn functions for split extensions while Theorem E describes the growth of Dehn functions for compactly presented groups with details. Even if the present contribution is quite long to read, the exposition of the material is very fluid so the reader can easily understand the main ideas and follow the principal techniques of proofs without problems.