RESEARCH STATEMENT COREY BREGMAN I study geometric group theory, which aims to uncover the relation between algebraic and geometric properties of groups. More specifically, I focus on the geometry of CAT(0) spaces, Culler-Vogtmann outer space, and K¨ahlergroups. The following are some of my ongoing and future research projects. • Studying the corank of special groups and the classification of special cube complexes with abelian hyperplanes (x1). • Understanding automorphisms of right-angled Artin groups and the Nielsen realization problem for finite subgroups (x2). • Describing the period mapping on outer space, with a view towards group cohomology (x3). • Classifying K¨ahlergroups that are abelian-by-surface or surface-by-surface extensions (x4). Special NPC cube complexes. Recently, the geometry of CAT(0) cube complexes featured prominently in Agol's solution [1] of two long-standing conjectures of Thurston in low-dimensional topology: the virtually Haken and virtually fibered conjecture for hyperbolic 3-manifolds. An essential ingredient of Agol's proof is that every hyperbolic 3-manifold group is virtually special, i.e. the fundamental group of a special CAT(0) cube complex. Special groups comprise a large class n containing many well-known families, such as free groups Fn, free abelian groups Z , and orientable surface groups. I defined a geometric invariant of special cube complexes called the genus (cf. x1) which generalizes the classical genus of a closed, orientable surface. I then show that having genus one characterizes all X with abelian fundamental group. Automorphisms of right-angled Artin groups (raags). Raags comprise a family of special n groups which interpolate between Fn and Z : they are defined by choosing a finite set of generators and specifying which generators commute. Passing to outer automorphism groups, raags give a n n unified framework for studying the relationship between Out(Fn) and GL(Z ) = Out(Z ). Given a raag A, Charney{Stambaugh{Vogtmann define a contractible simplicial complex on which a subgroup of Out(A) acts geometrically. Using this action, I give a geometric proof that the Torelli subgroup (cf. x2) I(A) ≤ Out(A) is torsion-free. The period mapping on outer space CVn. Culler{Vogtmann define a space CVn on which Out(Fn) acts properly. CVn consists of metric graphs Γ, together with an identification of π1(Γ) with Fn. The metric allows one to define a positive definite quadratic form on H1(Γ), giving a continuous map Φ from CVn to the space of rank n positive definite quadratic forms. The period mapping Φ is defined by analogy with the classical period map for complex algebraic curves and coincides with the period map in 1-dimensional tropical geometry. Understanding the fibers of Φ is related to computing the cohomology of I(Fn) while understanding the image of Φ is related to n the classification of lattices in R . In joint work with Neil Fullarton, we give a complete geometric description of the fibers of Φ. K¨ahlergroups. K¨ahlermanifolds are complex manifolds together with a compatible symplectic structure. Among the simplest examples of K¨ahlermanifolds are closed Riemann surfaces and 2n-dimensional tori. A group is said to be K¨ahler if it is realized as the fundamental group of a compact K¨ahlermanifold. Relatively few explicit examples of K¨ahler groups are known, making the construction of new examples of interest to complex geometers and geometric group theorists alike. One might hope to build new examples of K¨ahlergroups as extensions of known ones. Using 1 2 COREY BREGMAN techniques from surface topology, Letao Zhang and I showed that extensions of abelian groups by surface groups are virtually products. 1. Special cube complexes Gromov first introduced nonpositively curved (NPC) cube complexes as a source of easily con- structible examples of CAT(0) spaces [21]. Cube complexes are constructed by gluing Euclidean n-cubes [−1; 1]n together along their faces by isometries. We call a cube complex X non-positively curved (NPC) if its universal cover is CAT(0). A consequence of the CAT(0) condition is that NPC cube complexes are aspherical, and therefore their geometry is intimately connected with their fun- damental groups. Special cube complexes were first introduced by Haglund and Wise in [22] as a particular type of cube complex whose hyperplanes exhibit controlled behavior. If G = π1(X) for a (compact) special cube complex X then G is called (compact) special. Among their many notable properties compact special groups are known to be: (1) Linear over Z [22], (2) Residually torsion-free nilpotent [22], (3) Either virtually abelian or large [35]. The last of these properties means that a finite index subgroup is either abelian or surjects onto a free group Fn with n ≥ 2. We define the corank of a group G to be the largest n such that G surjects onto Fn. Wise asked whether one could avoid passing to a finite index subgroup: Question 1.1. (Wise [35]) If G is special and non-abelian, is its corank at least 2? To address this question we introduced an invariant of special cube complexes called the genus. Any special cube complex X contains many embedded codimension-1 subcomplexes called hyperplanes. Definition 1.2. (Bregman [10]) The genus g(X) is the maximal number of disjoint hyperplanes whose union does not disconnect X. If G is a special group, then define ∼ g(G) := sup fg(X) j π1(X) = G; X specialg : If g(X) = n, then the corank of π1(X) is at least n. Using the genus, we give a strong geometric answer to Question 1.1: Theorem 1.3. (Bregman [10]) Let X be a special cube complex. Then (1) g(X) = 0 if and only if X is CAT(0). (2) g(X) = 1 if and only if π1(X) is abelian. Moreover, if π1(X) is abelian, then X admits a cube complex collapse onto a cubulated torus. In particular, if π1(X) is not abelian, then the corank of π1(X) must be at least 2. Our theorem is stronger than Wise's question in the following sense. If π1(X) is not abelian, then there exists a map of cube complexes from X onto a wedge of two circles representing the surjection π1(X) F2. Theorem 1.3 implies that if G is non-abelian with first Betti number b1(G) < 2, then G is not special. In particular, it follows from our theorem that there exist many fibered hyperbolic 3-manifold groups which are not special, but which are virtually special by Agol's theorem [1]. We also show that for surface groups, our notion of genus agrees with the classical definition. For free groups, free abelian groups, and surface groups, we show in [10] that the genus always equals the corank, and it is natural to wonder whether this is always the case. In future work we will calculate the genus of other families of special groups, such as right-angled Artin groups (cf. x2). Research Proposal. Wise [35] showed that special groups admit a quasiconvex hierarchy, i.e. they can be built from the trivial group by iterated amalgamation and HNN-extension along quasiconvex subgroups. Free groups, free abelian groups, and surface groups each have hierarchies where every quasiconvex subgroup is abelian. RESEARCH STATEMENT 3 Conjecture 1.4. (Wise, Conjecture 14.11 [35] ) Every group with an abelian quasiconvex hierarchy is either abelian, or a free-by-abelian or surface-by-abelian extension. Using the structure of genus 1 special cube complexes we developed, we propose to show that every special group with an abelian quasiconvex hierarchy is actually a free product of free abelian and surface groups. By a theorem of Rips (cf. [9]), this would imply that special groups with an abelian quasiconvex hierarchy are exactly the groups which admit free actions on R-trees. 2. Automorphisms of right-angled Artin groups Given a group G, its outer automorphism group Out(G) is defined as Aut(G)=Inn(G), where Inn(G) is the group of inner automorphisms induced by conjugation. The abelianization map G ! Gab induces a map Ψ : Out(G) ! Out(Gab). We define the Torelli subgroup to be I(G) := ker Ψ. If Γ = (V; E) is a finite simplicial graph, the associated right-angled Artin group (raag) AΓ is the group with presentation AΓ = V [v; w]; if v; w 2 V share an edge in Γ : Raags are the prototypical examples of special groups, and conversely, Haglund and Wise [22] showed that every special group embeds in some raag. When Γ has no edges, then AΓ is free, whereas if Γ is a complete graph, AΓ is free abelian. Because of this, raags are said to interpolate n between Fn and Z . Pushing this analogy further, we study how well the outer automorphism n group Out(AΓ) interpolates between Out(Fn) and Out(Z ) = GLn(Z). Outer space and Nielsen realization. Recently, Charney{Stambaugh{Vogtmann [16] intro- duced a contractible, finite dimensional simplicial complex KΓ on which a subgroup U(AΓ) ≤ Out(AΓ) acts geometrically. KΓ is useful for understanding the group structure Out(AΓ). As an example, let I(AΓ) denote the Torelli subgroup for AΓ. From results of Day [18] one knows that I(AΓ) ≤ U(AΓ). In [10], we used the action of I(AΓ) on KΓ to give a geometric proof of Theorem 2.1. I(AΓ) is torsion-free for all Γ. This theorem is due to Baumslag{Taylor [7] when AΓ is free and Wade [34] and Toinet [32] in the general case. However, all three proofs are algebraic in nature, while ours makes use of the space KΓ and the geometry of cube complexes.