The following review is borrowed from a referee report, up to a few very minor changes (the referee report being supposed to give an opinion, unlike a public review: thus the few appreciations have been removed). It purports to provide an external description of the paper, given the fact that MathSciNet has not considered the paper worth a review (publishing a report by a reviewer whose contribution was to copy the 4-line abstract and change "we" into "the authors" MR3668649).
Cornulier and Tessera give two invariants that completely determine whether a Lie group has an polynomially bounded or an exponentially large Dehn function. The Dehn function is an important invariant in geometric group theory. Among other things, it determines whether a group is hyperbolic, and the difference between an exponential and a polynomial Dehn function is very closely connected to the large-scale geometry of a group and the topology of its asymptotic cone.
The first invariant is the SOL obstruction, which is simply a homomorphism to a particular class of groups with exponential Dehn function. Gromov suggested that this might be the only obstruction, but this turns out not to be true; The authors consider a second obstruction, which they call the 2-homological obstruction, which also implies that a Lie group has an exponential Dehn function. This obstruction is related to the existence of central or hypercentral extensions with certain geometrical properties. A similar obstruction was used in the authors's J. of Topology paper MR3145147 in which they construct groups whose asymptotic cones have unusual topology.
The main result in the current paper is that these two obstructions are necessary and sufficient. That is, a Lie group without the SOL obstruction and the 2-homological obstruction has a polynomially bounded Dehn function. This does not completely solve the problem of finding Dehn functions of Lie groups (as Wenger has shown MR2783380, even nilpotent Lie groups can have complicated Dehn functions), but it's a big step forward most previous work dealt with relatively small classes of solvable or arithmetic groups. Recall that the Dehn function is an invariant of a finitely or compactly presented group which measures the difficulty of reducing a product of generators (a word) that represents the identity to the empty word by using the relators in the presentation.
In order to prove their theorem, Cornulier and Tessera have to give an "algorithm" to reduce arbitrary words in Lie groups avoiding their obstructions to the empty word. The method they use is unexpected. Very roughly speaking, they show that if an algebraic group over a sufficiently large ring of functions has a presentation using relations of a particular form, then that algebraic group, taken over a smaller ring, has a polynomially bounded Dehn function. By building on results of Abels, they show that groups avoiding the two obstructions can be presented in a particular way (using amalgamation relators, tame relators, and welding relators), then, using combinatorial methods, they show that all of these relators have small filling areas.
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