THEGRADUATESTUDENTSECTION

WHATIS… 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

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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,

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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. These (homotopy classes of) homeomorphisms Theorem 1. The action of Mod(푆) on 풞(푆) is acylindrical. taken together form an inﬁnite subgroup of Mod(푆) which The usefulness of acylindricity is perhaps not imme- ﬁxes the vertex 훾 of 풞(푆), whence it is clear that the diately clear. The most productive setting for studying action of Mod(푆) on 풞(푆) cannot be proper. acylindricity is in the case where 푋 is a hyperbolic graph. How badly behaved is the action of Mod(푆) on 풞(푆)? This means that 푋 is a graph equipped with the graph Can something be said about it which is not a general metric, and the graph metric is (Gromov) hyperbolic. That statement about isometric group actions on graphs? is to say, there is a constant 훿 ≥ 0 such that for any It turns out that yes, indeed one can. The action is triple 푥, 푦, 푧 ∈ 푉(푋) of vertices and geodesic segments acylindrical, which is in some sense the next best thing [푥, 푦], [푦, 푧], [푥, 푧] ⊂ 푋, we have after properness. [푥, 푧] ⊂ 푁 ([푥, 푦] ∪ [푦, 푧]). Acylindrical actions were ﬁrst generally deﬁned by 훿 Bowditch [1] in 2008. Let 퐺 be a group acting by isometries In other words, every geodesic triangle is 훿–thin in the on a path-metric space 푋. The action of 퐺 on 푋 is sense that a 훿–neighborhood of two sides contains the acylindrical if for all 푟 ≥ 0, there exist constants 푅, 푁 ≥ 0 third side (see Figure 5). such that for any pair 푎, 푏 ∈ 푋 with 푑(푎, 푏) ≥ 푅, we have Note that the deﬁnition of hyperbolicity makes sense for any geodesic metric space, and indeed this is the |{푔 ∈ 퐺 ∣ 푑(푔 ⋅ 푎, 푎) ≤ 푟 and 푑(푔 ⋅ 푏, 푏) ≤ 푟}| ≤ 푁. deﬁnition of a hyperbolic (metric) space which is not The set of elements necessarily a graph. It is highly nonobvious though true that 풞(푆) is a hyperbolic graph, by a deep result of {푔 ∈ 퐺 ∣ 푑(푔 ⋅ 푎, 푎) ≤ 푟} H. Masur and Y. Minsky from 1999. is not quite the stabilizer of 푎, but rather the 푟-quasi- In the case of an action of a group 퐺 on a hyperbolic stabilizer of 푎. Acylindricity can be summed up as saying graph 푋, acylindricity of the 퐺-action gives a tractable that “for all 푟, simultaneous 푟-quasi-stabilizers of suf- geometric shadow of 퐺 in 푋, given by considering the ﬁciently distant points are uniformly small.” In more orbit of an arbitrary vertex 푣 ∈ 푉(푋). To make sense of informal terms, an acylindrical action is “uniformly this notion, we deﬁne the translation length of an element

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푔 ∈ 퐺, a deﬁnition which makes sense for any isometric acylindricity of the Mod(푆) action on 풞(푆) again inter- action of 퐺 on 푋. We write cedes to furnish a profusion of them, by the following 푑(푔푛 ⋅ 푥, 푥) recent result of F. Dahmani, V. Guirardel, and D. Osin: 휏(푔) = lim , 푛→∞ 푛 Theorem 3 (Dahmani–Guirardel–Osin (2017)). Let 퐺 be a a limit which always exists and which is independent of group acting acylindrically on a hyperbolic space 푋. Then the choice of 푥. there exists a natural number 푁 > 0 such that for every The translation length of 푔 is either positive or zero. loxodromic 푔 ∈ 퐺, the normal closure ⟨⟨푔푁⟩⟩ is free and In the former case, the element 푔 is called loxodromic. purely loxodromic. An example of a loxodromic isometry is a homothetic expansion of the upper half-space model for hyperbolic In recent years, there has been an explosion of re- space. One way that 휏(푔) can be zero is if some (or sults by many authors on acylindrical actions of various indeed every) 푔-orbit has ﬁnite diameter, in which case groups on hyperbolic spaces. In addition to mapping 푔 is called elliptic. An example of this latter case is class groups, examples of groups admitting nonelemen- rotation of hyperbolic space. General actions can have tary acylindrical actions on hyperbolic spaces include elements such that {푔푛 ⋅ 푥} is unbounded but where nonelementary hyperbolic groups, groups which are 푛∈ℤ nonelementary hyperbolic relative to proper subgroups, 푑(푔푛 ⋅ 푥, 푥) grows strictly sublinearly as a function of 푛, outer automorphism groups of free groups, Cremona in which case 푔 is parabolic. The map (푥, 푦) ↦ (푥 + 1, 푦) groups, nonvirtually nilpotent groups acting properly on of the upper half-plane is an example of a parabolic a hyperbolic space of uniformly bounded geometry, right- isometry of the upper half-space model of hyperbolic angled Artin groups, and compact 3-manifold groups space. Parabolic isometries can imbue group actions with which are not Seifert ﬁbered. A lot of these examples have signiﬁcant complexity, as in the case of lattices acting been organized and their properties developed recently on symmetric spaces. In many higher rank situations, by the work of J. Behrstock, M. Hagen, and A. Sisto, to parabolic elements can generate lattices which can often form the class of hierarchically hyperbolic groups. be shown to never admit interesting acylindrical actions. Acylindricity has been an extremely productive and For acylindrical actions on hyperbolic graphs, Bowditch pervasive concept in geometric group theory, and has led proved the following general fact which simpliﬁes the to fast paced and dramatic advances. Undoubtedly, it will picture somewhat: continue to do so for some time. Theorem 2. Let 퐺 be a group acting acylindrically on a hyperbolic graph 푋. Then every nontrivial 푔 ∈ 퐺 is References either loxodromic or elliptic. Moreover, there is a con- 1. Brian H. Bowditch, Tight geodesics in the curve stant 휖 > 0 depending only on the acylindricity and complex, Invent. Math. 171 (2008), no. 2, 281–300. hyperbolicity constants such that if 푔 is loxodromic then MR 2367021 (2008m:57040) 휏(푔) ≥ 휖. Image Credits In terms of terminology, acylindrical actions are either elementary or nonelementary. An action is elementary Figures courtesy of Thomas Koberda. if it is purely elliptic or if there is (essentially) only Photo of Thomas Koberda courtesy of Angelo Mao. one cyclic subgroup consisting of loxodromic elements. Nonelementary acylindrical actions are the only interest- ing ones. As might be expected, the Mod(푆) action on ABOUT THE AUTHOR 풞(푆) is nonelementary. It is possible to show that loxodromic elements are In addition to doing mathematics, exactly the mapping classes such that no power ﬁxes Thomas enjoys reading, foreign a simple closed curve, as was done by Thurston. Such languages, cooking, and running. mapping classes are called pseudo-Anosov, and are ar- guably the most interesting mapping classes. Thurston’s beautiful 1988 article in the Bulletin of the American Math- ematical Society provides an accessible introduction. The literature on the coarse geometry of subgroup of Mod(푆) Thomas Koberda which consist entirely of pseudo-Anosov elements (i.e. purely pseudo-Anosov subgroups) is vast, and is the do- main of convex cocompact subgroups of Mod(푆). Convex cocompact subgroups, which we will not deﬁne precisely here, are of central importance in the geometry of the moduli space of curves, hyperbolic group extensions, and the algebraic and geometric structure of mapping class groups. Note that it is not obvious a priori that there exist noncyclic purely pseudo-Anosov subgroups of Mod(푆). They can be produced directly with some work, but the

34 Notices of the AMS Volume 65, Number 1