Matthew Cordes

We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Björklund and the second-named author. Our theory is based on the notion of a quasi-isometric quasi-action (qiqac) of an approximate group on a metric space.

More specifically, we introduce a geometric notion of finite generation for approximate group and prove that every geometrically finitely-generated approximate group admits a geometric qiqac on a proper geodesic metric space. We then show that all such spaces are quasi-isometric, hence can be used to associate a canonical QI type with every geometrically finitely-generated approximate group. This in turn allows us to define geometric invariants of approximate groups using QI invariants of metric spaces. Among the invariants we consider are asymptotic dimension, finiteness properties, numbers of ends and growth type.

A particular focus is on qiqacs on hyperbolic spaces. Our strongest results are obtained for approximate groups which admit a geometric qiqac on a proper geodesic hyperbolic space. For such "hyperbolic approximate groups'' we establish a number of fundamental properties in analogy with the case of hyperbolic groups. For example, we show that their asymptotic dimension is one larger than the topological dimension of their Gromov boundary and that - under some mild assumption of being "non-elementary'' - they have exponential growth and act minimally on their Gromov boundary. We also show that every non-elementary hyperbolic approximate group of asymptotic dimension 1 is quasi-isometric to a finitely-generated non-abelian free group.

We also study proper qiqacs of approximate group on hyperbolic spaces, which are not necessarily cobounded. Here one focus is on convex cocompact qiqacs, for which we provide several geometric characterizations à la Swenson. In particular we show that all limit points of a convex cocompact qiqac are conical, whereas in general the conical limit set contains additional information.

Finally, using the theory of Morse boundaries, we extend some of our results concerning qiqacs on hyperbolic spaces to qiqacs on proper geodesic metric spaces with non-trivial Morse boundary.

Throughout this book we emphasize different ways in which definitions and results from geometric group theory can be extended to approximate groups. In cases where there is more than one way to extend a given definition, the relations between different possible generalizations are carefully explored.

We prove that Morse local-to-global groups grow exponentially faster than their infinite index stable subgroups. This generalizes a result of Dahmani, Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groups to a broad class of groups that contains the mapping class group, CAT(0) groups, and the fundamental groups of closed 3-manifolds. To accomplish this, we develop a theory of automatic structures on Morse geodesics in Morse local-to-global groups. Other applications of these automatic structures include a description of stable subgroups in terms of regular languages, rationality of the growth of stable subgroups, density in the Morse boundary of the attracting fixed points of Morse elements, and containment of the Morse boundary inside the limit set of any infinite normal subgroup.

We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic.

We study direct limits of embedded Cantor sets and embedded Sierpiński curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called \(\omega\)-Cantor spaces, and similarly, all limits of Sierpiński curves give homeomorphic spaces, called to \(\omega\)-Sierpiński curves. We then show that the former occur naturally as Morse boundaries of right-angled Artin groups, while the latter occur as Morse boundaries of fundamental groups of finite-volume, cusped hyperbolic 3-manifolds.

Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon–Thurston maps do not exist for infinite torsion-free normal hyperbolic subgroups of a non-hyperbolic CAT(0) group with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston maps do not exist for infinite torsion-free infinite-index normal CAT(0) subgroups with isolated flats in a non-hyperbolic CAT(0) group with isolated flats. We determine the isomorphism types of the normal subgroups in the above two settings.

The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic-like behavior in the space. A key property of this boundary is that a quasi-isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper we investigate when the converse holds. We prove that for \(X,Y\) proper, cocompact spaces, a homeomorphism between their Morse boundaries is induced by a quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.

In this paper we survey many of the known results about Morse boundaries and stability.

We build quasi-isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors.

We prove that, given any finite collection of finitely generated groups \(\mathcal{H}\) each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi-isometry types of one-ended groups which are hyperbolic relative to \(\mathcal{H}\).

A Kleinian group \(\Gamma < \mathrm{Isom}(\mathbb H^3)\) is called convex cocompact if any orbit of \(\Gamma\) in \(\mathbb H^3\) is quasiconvex or, equivalently, \(\Gamma\) acts cocompactly on the convex hull of its limit set in \(\partial \mathbb H^3\).

Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups.

Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.

Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalize these strategies by viewing any geodesic metric space as a countable union of stable subspaces: we show that every stable subgroup is a quasi-convex subset of a set in this collection and that the Morse boundary is recovered as the direct limit of the usual Gromov boundaries of these hyperbolic subspaces.

We use this approach, together with results of Leininger–Schleimer, to deduce that there is no purely geometric obstruction to the existence of a non-virtually-free convex cocompact subgroup of a mapping class group.

In addition, we define two new quasi-isometry invariant notions of dimension: the stable dimension, which measures the maximal asymptotic dimension of a stable subset; and the Morse capacity dimension, which naturally generalises Buyalo's capacity dimension for boundaries of hyperbolic spaces.

We prove that every stable subset of a right-angled Artin group is quasi-isometric to a tree; and that the stable dimension of a mapping class group is bounded from above by a multiple of the complexity of the surface. In the case of relatively hyperbolic groups we show that finite stable dimension is inherited from peripheral subgroups.

Finally, we show that all classical small cancellation groups and certain Gromov monster groups have stable dimension at most 2.

We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper \(\mathrm{CAT}(0)\) space this boundary is the contracting boundary of Charney and Sultan, and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichmüller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichmüller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichmüller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichmüller space into the Thurston compactification of Teichmüller space by projective measured foliations.

We study random nilpotent groups in the well-established style of random groups, by choosing relators uniformly among freely reduced words of (nearly) equal length and letting the length tend to infinity. Whereas random groups are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients of a free nilpotent group. This model reveals new phenomena because nilpotent groups are not "visible" in the standard model of random groups (due to the sharp phase transition from infinite hyperbolic to trivial groups).