SPECIAL FEATURE: INTRODUCTION INTRODUCTION SPECIAL FEATURE: Quantitative geometry Assaf Naor1 on at least two generators (5), and Courant Institute, New York University, New York, NY 10012 pffiffiffiffiffiffiffiffiffiffi DGðnÞ ≍ log n if G is the Heisenberg group (8). Similar questions with Hilbert Rather than describing a traditional field of word metric in a bi-Lipschitz way. One is space replaced by L1 are of fundamental mathematics, “quantitative geometry” refers therefore led to consider groups as metric importance to theoretical computer science to a modern way of thinking about geometry spaces that are indistinguishable under bi- (9). Another way to quantify the extent and its applications that occur in a wide Lipschitz deformations. This viewpoint is of to which a group G that is generated by fi range of mathematical disciplines. The over- monumental importance to group theory a nite symmetric set S fails to embed bi- arching theme here is that certain geometric and geometry, Gromov’s characterization Lipschitzly into Hilbert space was introduced fi problems can only be formulated after the of groups of polynomial growth (1) being by Guentner and Kaminker (10): they de ne introduction of auxiliary quantitative param- a seminal work in this area (see also ref. 2). the Hilbert compression exponent of G, α eters and asking for their asymptotic or ap- Thethirdarticle(3)inthepresentvolume denoted ðGÞ, to be the supremum over α ∈ ; proximate behavior. This leads to meaningful (to be described below) answers a question of those ½0 1 for which there exists a : → ℓ questions that do not necessarily have coun- Gromov related to his polynomial growth Lipschitz mapping f G 2 satisfying − ≥ ; α ; ∈ terparts in classical “qualitative” geometric theorem, and the first article (4) in this vol- jjf ðxÞ f ðyÞjj2 cdSðx yÞ for every x y G. · ; · investigations, a viewpoint that allows one ume also resolves a basic question in geo- Here dSð Þ denotes the word metric in- ∈ ; ∞ to uncover deep and useful structural infor- metric group theory, exhibiting for the first duced by S and c ð0 Þ may depend on ; ; α ; mation that appears only in the quantitative time that two natural quantitative geometric G f but not on x y. Again, being in- fi fi regime. Quantitative reasoning allows one parameters of groups (homological and dependent of the speci c choice of a nite to tame complicated objects and phenom- homotopical Dehn functions) can have generating set, αðGÞ is an algebraic invariant ena that are unwieldy if one insists on exact different asymptotics for the same group. of G, yet there are only a few known meth- measurements. It is often the case that a rich Whether or not a given finitely generated ods to compute it, some of which involve yet structured picture emerges only if one group admits a geometrically faithful em- interesting links to harmonic analysis and asks the correct questions by allowing for bedding into a certain well-behaved space, probability theory. Note that both the controlled errors. say, into Hilbert space or into an L1ðμÞ space, asymptotics of DGðnÞ and the Hilbert The term quantitative geometry was coined is a basic question in geometric group theory, compression exponent αðGÞ are quantita- as an attempt to formulate the widespread with many important algebraic and topo- tive invariants that encode structural in- understanding of many researchers world- logical implications (see ref. 5 for a striking formation about the group G, yet they do wide that in recent years numerous natural example). Here the notion of a “geometrically not have qualitative counterparts. questions have emerged that mandate the faithful” embedding can take several mean- study of geometric problems from an ap- ings; there is, for example, great interest in bi- Geometric Topology. Topology is another proximate perspective. This trend is some- Lipschitz and coarse embeddings. Focusing source of a wealth of important quantitative times dictated by application areas in which on the bi-Lipschitz case, it is an intriguing geometric questions. Despite the numerous exact computation is infeasible yet approxi- and beautiful conjecture (6) that a finitely remarkable achievements of algebraic topol- fi mate measurements are of crucial impor- generated group G admits a bi-Lipschitz ogy, the dif culty of understanding classical tance, but it arises most often as a natural embedding into Hilbert space if and only if qualitative topological problems quantita- modern development of traditional mathe- G has an Abelian subgroup of finite index. tively has been emphasized by Gromov (11). matical disciplines, where quantitative refine- Therefore, to classify non-virtually-Abe- For example, suppose that X; Y are finite ments of classical investigations are necessary lian groups G in terms of their embeddability complexes and Y is simply connected. If to probe deeper and fully understand basic into Hilbert space, one must quantify the f ; g : X → Y are homotopic “well-behaved” mathematical objects. extent to which G does not admit a bi-Lip- maps, are they also homotopic through maps that are equally well behaved? There What Is Quantitative Geometry? schitz embedding into Hilbert space. There is more than one natural (and fruitful) way to are many ways to interpret what well be- Rather than attempting to give formal defi- do this. For every n ∈ N,onecanaskwhat haved means in this general context. Here nitions, we shall illustrate the above discus- is the largest Hilbertian bi-Lipschitz dis- we take this term to mean that the map- sion through examples from various areas tortion of an n-point subset of G,and pings in question are Lipschitz. Therefore, of mathematics. ; : → denoting this number by DGðnÞ,in-supposing that f g X Y are homotopic Geometric Group Theory. AgroupG that vestigate the rate at which DGðnÞ tends to L-Lipschitz functions, can they be homo- is generated by a finite symmetric set S infinity.Thisratedoesnotdependon toped through KL-Lipschitz functions, or becomes a geometric object if one endows aspecific finite generating set, so it is perhaps one can even find a homotopy it with the word metric that is induced by S. a genuine algebraic invariant of G. It turns that is itself KL-Lipschitz? Here K is If one nevertheless wants to study groups as out to be quite difficult to evaluate the as- Author contributions: A.N. wrote the paper. algebraic entities, the generating set itself is ymptotic behavior of DGðnÞ, and this has of minor or no importance, but replacing been computedpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in only a few cases, e.g., The author declares no conflict of interest. 1 S by another generating set T changes the DGðnÞ ≍ log log n if G is a free group E-mail: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1320388110 PNAS Early Edition | 1of4 Downloaded by guest on September 27, 2021 allowed to depend on X; Y but not on topological spaces, a lot of structure remains problems that are efficiently solvable, and f ; g; L. The seventh article (12) of the if one insists that homeomorphisms are the resulting “rounding problem” asks for present volume treats this question by quantitatively continuous. For example, an efficient way to generate a discrete solu- obtaining a necessary and sufficient con- since for distinct p; q ∈ ½1; ∞Þ the se- tion from a continuous solution while losing a dition for the existence of a KL-Lipschitz quence spaces ℓp and ℓq do not have the (hopefully small) definite multiplicative fac- homotopy but also shows that sometimes same isomorphic local linear structure, it tor. Conversely, through mechanisms such as one cannot find Lipschitz homotopies despite follows from Ribe’s theorem that ℓp and ℓq Khot’s Unique Games Conjecture (26), one the fact that homotopies through Lipschitz are not uniformly homeomorphic. can relate the computational complexity of functions are possible. We have indicated just Note a crucial feature of Ribe’s theorem: certain optimization problems and specifi- one example of quantitative geometric its conclusion involves an unspecified con- cally the “hardness threshold,” above which refinements of topological concepts and stant K, so it can be used only if one con- it becomes feasible to find an approximate questions, but the literature contains many siders properties of Banach spaces that are solution to continuous quantitative geo- investigations (and many open questions) of insensitive to deformations that result in metric questions; see ref. 22 for more on this this type, all of which belong under the a loss of such a constant factor. Moreover, topic. In graph theory, one sometimes quantitative geometry rubric. the conclusion of Ribe’s theorem involves relates geometric invariants to combinato- properties of finite dimensional subspaces rial graphs [e.g., the Lovász theta function Differential Geometry. Geometric flows (a.k.a. local properties), and since all of the (28)], leading naturally to interesting geo- and limiting constructions such as blow-ups norms on Rn are equivalent, finite di- metric problems of a quantitative/asymp- and tangent cones are commonly studied in mensional normed spaces become distin- totic nature (29). The fourth article (30) in differential geometry, and it is very natural guishable only if one studies them in the the present volume contains results along (and useful) to refine these investigations quantitative regime. This point of view these lines. As a last example, we consider quantitatively. The fifth article (13) of the gives rise to the local theory of Banach spaces: the area of incidence geometry. Classically, present volume describes interesting con- a deep and rich theory (18, 19) that over the one is interested here in understanding the tributions to this aspect of quantitative ge- last five decades has had a transformative constraints that are imposed on a finite set of ometry.