WHAT IS ... an acylindrical group action?

The group Z acts on the real line R by translation. It is difficult to find a nontrivial group action which is easier to understand: the orbit of every point moves off to infinity at a steady and predictable rate, and the group action preserves the usual Euclidean metric on R. Of course, this action is a covering space action, and the quotient space of the action is the circle, which is completely free of any topological pathologies. Regular covering spaces in algebraic topology give rise to prototypically nice group actions. Among the most important features of a deck group action on a covering space is that it is free (i.e. no nontrivial element of the deck group has a fixed point) and properly discontinuous (i.e. for every compact subset K of the cover, there are at most finitely many deck group elements g such that g ̈ K X K ‰ ∅, at least in the case where the base space is locally compact). For certain purposes in topology and geometry, one can relax the freeness of an action without introducing insurmountable difficulties. Discrete groups of isometries of hyperbolic space, for example, oftentimes contain torsion elements such as rotations, and finite order isometries of Euclidean or hyperbolic spaces always have a fixed point. By considering quotients of Euclidean and hyperbolic spaces by discrete groups of isometries, one naturally obtains the class of Euclidean and hyperbolic orbifolds, thus enlarging the class of Euclidean and hyperbolic manifolds. Orbifolds enjoy many of the salient features of manifolds, so that a mild relaxation of freeness of group actions still allows for reasonable geometry to persist. Relaxing proper discontinuity can lead to some pathological phenomena, for instance quotient maps whose quotient topologies fail to be Hausdorff or even fail to have any nontrivial open sets. Consider, for example, a group of rotations of the circle generated by an irrational multiple of π. Since the circle is compact, this action of Z cannot be properly discontinuous – indeed, every orbit is countably infinite and dense. Hence, the quotient is an uncountable space with no open sets except the empty set and the whole space. Nevertheless, group actions which are not properly discontinuous abound in mathematics and have led to the development of entire fields, such as noncommutative geometry in the sense of A. Connes. Group actions which are not properly discontinuous are also important and common in geometric group theory, with the following example being of central importance: Let S be an orientable surface and let γ Ă S be a simple closed curve, as illustrated in Figure 1 or in Figure 2. A curve is essential if it is not contractible to a point, and nonperipheral if it is not homotopic to a puncture or boundary component of S . The curve graph of S , denoted C pS q, is the graph whose vertices are nontrivial homotopy classes of essential, nonperipheral, simple closed curves, and whose edge relation is given by disjoint realization. That is, γ1 and γ2 are adjacent if they admit representatives which are disjoint. Thus, the curve graph encodes the combinatorial topology of one–dimensional submanifolds of S . Note that for relatively simple surfaces, C pS q may be empty or may fail to have any edges as they are defined here. For sufficiently complicated surfaces however, C pS q has very intricate and interesting structure. Whereas the curve graph as defined here is a manifestly combinatorial object, it is also a geometric object with the metric being given by the graph metric. It is an interesting exercise for the reader to prove that if C pS q admits at least one edge, then C pS q is connected, is locally infinite, and has infinite diameter. The mapping class group of S is the group of homotopy classes of orientation preserving homeomorphisms of S , and is written ModpS q. Mapping class groups are of central interest to geometric group theorists, as well as of significant interest to algebraic geometers, to topologists, and to homotopy theorists. From the point of view of geometric group theory, mapping class groups are studied via the geometric objects on which