THE GRADUATE STUDENT SECTION WHAT IS… an Acylindrical Group Action? Thomas Koberda Communicated by Cesar Silva The group ℤ acts on the real line ℝ by translation. It is for example, oftentimes contain torsion elements such as difficult to find a nontrivial group action which is easier to rotations, and finite order isometries of Euclidean or hy- understand: the orbit of every point moves off to infinity perbolic spaces always have a fixed point. By considering at a steady and predictable rate, and the group action quotients of Euclidean and hyperbolic spaces by discrete preserves the usual Euclidean metric on ℝ. Of course, this groups of isometries, one naturally obtains the class of action is a covering space action, and the quotient space Euclidean and hyperbolic orbifolds, thus enlarging the of the action is the circle, which is completely free of any class of Euclidean and hyperbolic manifolds. Orbifolds topological pathologies. enjoy many of the salient features of manifolds, so that Regular covering a mild relaxation of freeness of group actions still allows spaces in algebraic for reasonable geometry to persist. topology give rise One can relax the Relaxing proper discontinuity can lead to some patho- to prototypically nice logical phenomena, for instance quotient maps whose group actions. Among freeness of an quotient topologies fail to be Hausdorff or even fail to the most important action without have any nontrivial open sets. Consider, for example, a features of a deck group of rotations of the circle generated by an irrational group action on a cov- introducing multiple of 휋. Since the circle is compact, this action of ering space is that it ℤ cannot be properly discontinuous—indeed, every orbit is free (i.e. no non- insurmountable is countably infinite and dense. Hence, the quotient is an trivial element of the uncountable space with no open sets except the empty set deck group has a fixed difficulties. and the whole space. Nevertheless, group actions which point) and properly are not properly discontinuous abound in mathematics discontinuous (i.e. for every compact subset 퐾 of the and have led to the development of entire fields, such cover, there are at most finitely many deck group ele- as noncommutative geometry in the sense of A. Connes. ments 푔 such that 푔⋅퐾∩퐾 ≠ ∅, at least in the case where Group actions which are not properly discontinuous are the base space is locally compact). also important and common in geometric group theory, For certain purposes in topology and geometry, one can with the following example being of central importance: relax the freeness of an action while keeping discreteness, Let 푆 be an orientable surface and let 훾 ⊂ 푆 be without introducing insurmountable difficulties. Discrete a simple closed curve, as illustrated in Figure 1 or in groups of isometries of Euclidean and hyperbolic space, Figure 2. A curve is essential if it is not contractible to a point, and nonperipheral if it is not homotopic to a Thomas Koberda is assistant professor of mathematics at the puncture or boundary component of 푆. The curve graph University of Virginia. His email address is thomas.koberda of 푆, denoted 풞(푆), is the graph whose vertices are @gmail.com. nontrivial homotopy classes of essential, nonperipheral, For permission to reprint this article, please contact: simple closed curves, and whose edge relation is given [email protected]. by disjoint realization. That is, 훾1 and 훾2 are adjacent if DOI: http://dx.doi.org/10.1090/noti1624 they admit representatives which are disjoint. Thus, the January 2018 Notices of the AMS 31 THE GRADUATE STUDENT SECTION Figure 3. The subgraph of 풞(푆) spanned by the curves in Figure 2. The colored curves are represented by vertices of the corresponding color. The graph metric distance from the purple curve to the red curve in 풞(푆) is exactly two. of 푆, and is written Mod(푆). Mapping class groups are of central interest to geometric group theorists, as well as of significant interest to algebraic geometers, topologists, and homotopy theorists. From the point of view of geo- metric group theory, mapping class groups are studied via the geometric objects on which they act. Homeomor- Figure 1. A surface of genus 5. The blue curve is phisms of 푆 act on the set of embedded loops on 푆, and essential and nonseparating. The green curve is similarly homotopy classes of homeomorphisms act on inessential. The purple curve bounds a puncture or homotopy classes of embedded loops on 푆, and hence boundary component, denoted by a red 푥, and is of simple closed curves. Since the adjacency relation in therefore peripheral. 풞(푆) is a topological property, Mod(푆) acts by graph automorphisms and hence by graph metric isometries on 풞(푆). As natural as the action of Mod(푆) on 풞(푆) is, its curve graph encodes the combinatorial topology of one- geometry is extremely complicated. For one, the quotient dimensional submanifolds of 푆. Note that for relatively 풞(푆)/ Mod(푆) is finite, since two simple closed curves 훾 simple surfaces, 풞(푆) may be empty or may fail to 1 and 훾 are in the same mapping class group orbit if and have any edges as they are defined here. For sufficiently 2 only if 푆\훾 and 푆\훾 are homeomorphic to each other, complicated surfaces however, 풞(푆) has a very intricate 1 2 as follows easily from the classification of surfaces. Thus, and interesting structure. the action of Mod(푆) on 풞(푆) is highly transitive. This is in spite of the fact that 풞(푆) is locally infinite, as mentioned above: if 풞(푆) has at least one edge, then each vertex of 풞(푆) has infinite degree. Thus, the action of Mod(푆) on 풞(푆) is far from properly discontinuous. Note that proper discontinuity (as we have defined it at least) is perhaps not the best property to require from the action, since 풞(푆) is not locally compact (by virtue of being a locally infinite graph). A better notion which is meaningful for actions on spaces like 풞(푆) is properness. If 퐺 is a group generated by a finite set 푆, then 퐺 can be viewed as a metric space by declaring 푔 and ℎ to have distance one if 푔 = ℎ ⋅ 푠 for some 푠 ∈ 푆, and in general defining the distance between 푔 and ℎ to be the least 푛 such that 푔 = ℎ ⋅ 푠1 ⋯ 푠푛 for elements {푠1, … , 푠푛} ⊂ 푆. The reader may recognize this as the graph metric on the (right) Cayley graph of 퐺 with respect to 푆. If 퐺 acts on Figure 2. A surface of genus 5 with four essential a metric space 푋, the action is proper if (roughly) for all curves drawn. The subgraph of 풞(푆) spanned by 푥 ∈ 푋, the orbit map 퐺 → 푋 given by 푔 ↦ 푔 ⋅ 푥 is a proper them is given in Figure 3. map of metric spaces. The first example considered in this article, i.e. the translation action of ℤ on ℝ, is a proper Whereas the curve graph as defined here is a manifestly action. Note that we can build another action of ℤ on ℝ, combinatorial object, it is also a geometric object with where a generator of ℤ acts by multiplication by 2. This the metric being given by the graph metric. action of ℤ on ℝ is not proper. Returning to the situation It is an interesting exercise for the reader to prove that at hand, since vertices of 풞(푆) have infinite stabilizers in if 풞(푆) admits at least one edge, then 풞(푆) is connected, Mod(푆), the action of Mod(푆) on 풞(푆) is not proper. is locally infinite, and has infinite diameter. One way to see this is to observe the following: let The mapping class group of 푆 is the group of homo- 훾 ⊂ 푆 be an essential, nonperipheral, simple closed curve topy classes of orientation preserving homeomorphisms as in Figure 4. The surface 푆\훾 is a surface with boundary, 32 Notices of the AMS Volume 65, Number 1 THE GRADUATE STUDENT SECTION Figure 5. A path-metric space (such as a connected Figure 4. By considering an essential, nonperipheral, graph) is hyperbolic if there exists a 훿 ≥ 0 such that simple closed curve in red on this surface of genus 2, for every geodesic triangle, a 훿-neighborhood of two we can see that the action of Mod(푆) on 풞(푆) cannot sides contains the third side. be proper. Fortunately, it satisfies a weaker property: it is acylindrical. proper on sufficiently distant pairs of points.” Drop- ping the uniformity condition (i.e. replacing the uniform albeit simpler (in the sense that the Euler characteristic constant 푁 by a requirement that the relevant subset of is strictly larger). 퐺 is finite), one gets the closely related notion of a weakly The surface 푆\훾 generally properly discontinuous action. This latter notion appears admits many homotopically The next best in a 2002 paper of M. Bestvina and K. Fujiwara. nontrivial homeomorphisms Observe that, like many concepts in geometric group which act by the identity near thing after theory and coarse geometry, acylindricity is blind to 훾, which therefore extend to phenomena on a bounded scale. For instance, a group homeomorphisms of 푆 which properness. action on a bounded metric space is always acylindrical: fix 훾. Moreover, one can build just let 푅 be greater than the diameter of 푋. the Dehn twist about 훾, which Bowditch proved the following fundamental result: is given by cutting 푆 open along 훾 and regluing with a full twist.