RESEARCH STATEMENT I Study Geometric Group Theory, Which Aims to Uncover the Relation Between Algebraic and Geometric Properties

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 Kählergroups. 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 (§1). • Understanding automorphisms of right-angled Artin groups and the Nielsen realization problem for finite subgroups (§2). • Describing the period mapping on outer space, with a view towards group cohomology (§3). • Classifying Kählergroups that are abelian-by-surface or surface-by-surface extensions (§4). 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. §1) 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. §2) 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. Kählergroups. Kählermanifolds are complex manifolds together with a compatible symplectic structure. Among the simplest examples of Kählermanifolds are closed Riemann surfaces and 2n-dimensional tori. A group is said to be Kähler if it is realized as the fundamental group of a compact Kählermanifold. Relatively few explicit examples of Kähler 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 Kählergroups 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 fundamental 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 {g(X) | π1(X) = G, X special} .

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. §2).

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 ∈ 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] introduced 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. Points in KΓ parametrize NPC cube complexes with fundamental group AΓ called blow-ups of Salvetti complexes. If φ ∈ U(AΓ) has prime order, then standard group cohomology implies that φ y KΓ has a ﬁxed point. This ﬁxed point corresponds to a blow-up together with an automorphism f realizing the automorphism φ. We then prove Theorem 2.2. (Bregman [10]) Let X be a blow-up of a Salvetti complex. Then any non-identity automorphism of X acts non-trivially on H1(X). The argument above is a special case of the Nielsen realization problem. In its strongest form, the Nielsen realization problem for Out(AΓ) asks

Problem 2.3. For each raag AΓ, produce a finite dimensional, contractible space XΓ satisfying (1) The points of XΓ parametrize NPC cube complexes with fundamental group AΓ. (2) Out(AΓ) acts on XΓ properly discontinuously. (3) The fixed set of any finite subgroup H ≤ Out(AΓ) is non-empty and contractible. Culler [17], Khramtsov [25], and Zimmermann [36] independently showed that every finite subgroup of Out(Fn) can be realized as an automorphism group of a marked graph, which, together with work of Krstić–Vogtmann [29] implies the solution of Problem 2.3 when AΓ is a free group. For general raags, Hensel and Kielak [23] show that given a finite subgroup H ≤ U(AΓ), there exists some cube complex X with π1(X) = AΓ whose automorphism group realizes H, but it is not clear whether these cube complexes can be parametrized as an XΓ. Such an XΓ would be a 4 COREY BREGMAN classifying space for proper actions, also known as an EG. The existence of such a space is also related to the Baum–Connes Conjecture for Out(AΓ) [6].

Research Proposal. We propose to construct an XΓ extending the KΓ of Charney–Stambaugh– Vogtmann. It may be necessary to consider general NPC polyhedral complexes instead of just NPC cube complexes. As a ﬁrst step, we will use our understanding of automorphisms of blow-ups to resolve Problem 2.3 in the restricted case where U(AΓ) = Out(AΓ), since in this case taking XΓ = KΓ already satisﬁes (1) and (2) of Problem 2.3.

Abstract commensurators. One measure of the internal symmetries of a group is its outer automorphism group. A classical result of Hua–Reiner [24] from number theory states that the outer automorphism group Out(GLn(Z)) is Z/2 or Z/2 × Z/2 for all n. On the other hand, Khramtsov [26] and Bridson–Vogtmann [15] showed that Out(Aut(Fn)) and Out(Out(Fn)) are each trivial, for n ≥ 3. In contrast, Fullarton and I showed that general raags exhibit starkly diﬀerent behavior. Theorem 2.4. (Bregman-Fullarton [11]) For each n ≥ 2, there exists an inﬁnite family of graphs

{Γi} such that Out(Out(AΓi )) contains PGLn(Z). A stronger measure of the symmetry of a group is its abstract commensurator, denoted Comm(G). This group is deﬁned as the group of equivalence classes of isomorphisms between ﬁnite index subgroups of G. The abstract commensurators of GLn(Z) and Out(Fn) have already been calculated: ∼ • Comm(GLn(Z)) = GLn(Q). ∼ • Comm(Out(Fn)) = Out(Fn), for n ≥ 4 (Farb–Handel [20]).

For the examples Γi in Theorem 2.4, we calculate that Comm(Out(Γi)) is virtually a product of linear groups over Q, as in the case of GLn(Z).

Research Proposal. Calculate the abstract commensurators of Out(AΓ) for general raags AΓ. In particular, an example of a raag with Out(AΓ) virtually free would provide an example with a large abstract commensurator which are distinct from the GLn(Q) examples.

3. The period mapping on outer space

Let Qn be the space of rank n, positive definite quadratic forms. Given a marked, metric graph Γ one can define a positive definite quadratic form qΓ on H1(Γ), and the association Γ 7→ qΓ defines a continuous map Φ : CVn → Qn called the period mapping. The period mapping Φ is a free group analog of the classical Abel–Jacobi map for closed Riemann surfaces. It has applications to n n classification of lattices Z ⊂ R [33], and more recently, 1-dimensional Tropical geometry [30], [31]. The classical Abel–Jacobi map on Teichmüllerspace associates to each marked Riemann surface its matrix of periods in the Siegel upper half plane. The map factors through Torelli space, and the induced map is a 2:1 branched cover, branched along the hyperelliptic locus. It is natural to wonder whether the analogous description holds for free groups. Let I(Fn) ≤ Out(Fn) denote the Torelli subgroup. There is an exact sequence

1 → I(Fn) → Out(Fn) → GLn(Z) → 1.

The map Φ factors through the quotient Tn := CVn/I(Fn). A graph Γ is hyperelliptic if it admits an involution acting as −I on H1(Γ). Denote the set of hyperelliptic graphs in Tn by Hn.

Question 3.1. Describe the ﬁbers of the period mapping on Tn. Is it a branched cover along Hn? Building on work of Baker [5], who computed the case n = 3, Fullarton and I showed that the period mapping is not in general a branched cover along Hn. However, the ﬁbers on Tn do possess some nice properties and we are able to describe them explicitly for all n: RESEARCH STATEMENT 5

Theorem 3.2. (Bregman–Fullarton [12]) Connected components of ﬁbers of the period mapping on Tn are aspherical and π1-injective, with fundamental group lying in the pure symmetric automorphism group.

Here, the pure symmetric automorphism group is the subgroup which takes each generator of Fn to a conjugate of itself. Tn is an Eilenberg-Maclane space for I(Fn), which is torsion-free and has cohomological dimension 2n−4. Bestvina–Bux–Margalit [8] showed that H2n−4(I(Fn)) is infinitely generated. In particular, when n = 3, this implies that I(Fn) is not finitely presented. For n > 3, whether I(Fn) is finitely presented is a major open problem. When k = 2, work of Day–Putman [19] shows that Hk(I(Fn)) is finitely generated as a GLn(Z)-module, but for k ≥ 3, this is still open.

Research Proposal. In continuing work with Neil Fullarton, we propose to use our understanding of the fibers of the period mapping to find non-trivial homology classes in Hk(I(Fn)) = Hk(Tn). In particular, does the homology of a fiber inject into the homology of Tn? For k ≥ 3, is Hk(Tn) finitely generated as a GLn(Z)-module?

Focusing on hyperelliptic graphs, we also gave a presentation for the centralizer HOut(Fn) of the hyperelliptic involution. Since I(Fn) is torsion-free, the fundamental group of Hn is just ST (n) := HOut(Fn) ∩ I(Fn). This group is not very well understood, even in the surface case.

Research Proposal. Is ST (n) finitely generated/presented? The case n = 3 provides an inter- esting and tractable starting point for two reasons. The first is that we can show H3 deformation retracts onto a 2-complex in T3. The second is the fact that Krstic–McCool [28] have shown that I(F3) is not finitely presented. It seems likely that a careful understanding of the spine of H3, possibly using PL-Morse theory techniques will allow us to compute ST (3).

4. Kahler¨ groups As we stated in the introduction, relatively few examples of Kählergroups are known. One might hope to build new examples of Kähler groups as extensions of known ones. Surface groups and even rank finitely generated abelian groups are Kähler,hence we are led to consider the following types of extensions. Let E be a finitely generated group, and suppose E fits into a short exact sequence

(1) 1 → Sg → E → A → 1, where Sg is the fundamental group of a closed surface of genus g ≥ 2, and A is abelian. Question 4.1. Under what conditions is E a Kählergroup? ± ± Note that any such E is determined by a homomorphism A → Modg , where Modg is the (unori- ented) mapping class group of genus g. Thus, understanding possible extensions is the same as ± understanding abelian subgroups of Modg . Since the product of a Riemann surface Σg and an 2n 2n 2n even dimensional torus T is Kähler,we see that E = π1(Σg × T ) = Sg × Z is Kähler. In forthcoming joint work with Letao Zhang we prove this is essentially the only possibility Theorem 4.2. (Bregman-Zhang [13]) Let E as in short exact sequence (1) be Kähler.Then there 0 0 ∼ 2n exists E ≤ E of finite-index such that E = Sg × Z . Conversely, if A is finitely generated abelian and has even rank, any homomorphism ρ : A → Modg with finite image gives a Kählerextension.

A Kählergroup is fibered if it admits a surjection onto a surface group Sh with h ≥ 2. The reason for this terminology comes from a result of Catanese (cf. [2]) which states that if X is compact Kähler,then X admits a holomorphic surjection onto a Riemann surface of genus h0 ≥ h with connected fibers if and only if π1(X) surjects onto Sh . If one replaces A in sequence (1) with Sh for h ≥ 2, one can ask what possible fibered Kählergroups can occur. Examples of complex surfaces due to Kodaira [27] and Atiyah [4] demonstrate that E need not be a virtual product, and 6 COREY BREGMAN in this case the surface will be of general type.

Research Proposal. In joint work with Letao Zhang, we propose to study extensions of Sh by Sg where g, h ≥ 2. We would like a description of the possible monodromies of such actions. This would extend the program initiated by Arapura [3] for projective and quasi-projective varieties to all compact Kählergroups. Understanding surface group extensions is a first step towards classifying fibered Kählergroups and understanding complex projective surfaces of general type.

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Department of Mathematics, Rice University, 6100 Main Street, Houston, TX 77005, U.S.A. E-mail address: [email protected]