Research statement Harry Petyt

1. Research setting My research is in the area of geometric theory, which uses geometric tools to study infinite discrete groups. In particular, I am interested in using metric notions of nonpositive curvature to understand groups. The original motivation for the study of groups admitting a finite presentation came from Poincaré’s introduction of the fundamental group for closed manifolds [Poi95]. Tietze showed that such fundamental groups are always finitely presented [Tie08], and conversely it can be seen that in fact every finitely presented group is the fundamental group of a closed 4–manifold. Much early work in the area was driven by Dehn’s formulation [Deh10, Deh11] of the word, conjugacy, and isomorphism problems for finitely presented groups [CM82], but by the middle of the 20thcentury, results such as the examples of Novikov and Boone and the remarkable Adian–Rabin theorem had shown that even some of the most basic questions about finitely presented groups cannot be answered in full generality. Thus any reasonable theory should make additional assumptions beyond finite presentability. There followed a return to the geometric methods of Dehn [LS77], and a revolutionary step was taken by Gromov with his introduction of hyperbolic groups [Gro87]. One can view a group as a by identifying it with one of its Cayley graphs. Hyperbolic groups are defined by a metric negative curvature condition that (does not depend on the choice of and) has strong group-theoretic consequences. They have been studied intensively since their introduction and are now well understood. However, whilst in a sense almost all finitely presented groups are hyperbolic [Gro87, Ol’92], there are many naturally-arising groups with hyperbolic-like features that are not hyperbolic. Indeed, the strict negative curvature condition means that Z2 cannot be a subgroup of any . In recent years, a very active area of research has been to study finitely presented groups that are defined by some weaker form of nonpositive curvature. Examples include relative hyperbolicity, CAT(0) geometry, and semihyperbolicity. A more recent promising notion is that of hierarchical hyperbolicity, which has received a lot of attention from many authors since its introduction [BHS17a, DHS17, Hae20, HS20, Mou19, ABD21, Vok17, Spr17, RST18, BR20a, ANS19, CDG20, HHL20, BR20b, DDLS20, Mil20, DMS20, Hug21]. Hierarchical hyperbolicity. There are several key families of groups that one would like to understand by nonpositive curvature techniques, and a principal one is that of the mapping class groups of (compact, orientable) surfaces. The mapping class group of a surface is the group of isotopy classes of homeomorphisms of that surface. These classically- studied groups already appeared in the work of Dehn, Nielsen, and others a century ago, and have been fundamental to the study of 3–manifolds since the geometric insights of Thurston [Thu88]. Although mapping class groups have free abelian subgroups and thus are not hyperbolic in all but the simplest cases, seminal work of Masur–Minsky established that they do have some striking features of “local hyperbolicity” [MM99, MM00]. Another fundamental class of examples is provided by CAT(0) cube complexes, intro- duced by Gromov in [Gro87]. A cube complex is just a cell complex whose cells are Euclidean cubes. A cube complex is CAT(0) if it is simply connected and satisfies a local nonpositive curvature condition that is very robust, in the sense that it arises naturally in several metric settings [Che00, Lea13, Mie14]. CAT(0) cube complexes have been a very active area of research recently, in part thanks to the role they played in the ma- jor advances in 3–manifold theory due to Agol, Wise, and others [Ago13, Wis21]. It has been shown that most (conjecturally all) groups that admit a geometric on a CAT(0) cube complex enjoy similar “local hyperbolicity” features to mapping class groups [BHS17b, HS20]. This common structure has been formalised by Behrstock–Hagen–Sisto [BHS17b, BHS19] to form the class of hierarchically hyperbolic groups (a.k.a. HHGs). There are many exam- ples of HHGs other than mapping class groups and groups acting geometrically on CAT(0) 1 Research statement Harry Petyt cube complexes, such as most 3–manifold groups [HRSS], extra-large type Artin groups [HMS], the genus two handlebody group [Mil20], and various extensions and quotients of mapping class groups [BHS17a, BHMS20, DDLS20, Rus21], as well as a number of group- theoretic combinations of these [RS20, BR20a, BR20b]. Furthermore, being an HHG has some strong consequences, such as finite asymptotic dimension [BHS17a], Tits’ alternative [DHS17], control of quasiflats [BHS21], and so forth, as well as those given by some of the contributions described below. My research so far has centred around hierarchical hyperbolicity: relating it to other notions of nonpositive curvature, studying the large-scale geometry it engenders, and es- tablishing structural results for the class.

2. Hierarchical hyperbolicity and injective metric spaces Injective (also known as hyperconvex) metric spaces were introduced in the 50s [AP56]. They are defined by a ball-intersection property reminiscent of the classical Helly prop- erty studied in combinatorics and discrete geometry (though the name comes from their alternative characterisation as the injective objects in the category of metric spaces with distance–non-increasing maps), and have good metric properties similar to those of CAT(0) spaces [Isb64, Dre84, Lan13, BH99]. In joint work with Haettel and Hoda, we showed how this theory can be applied to the study of hierarchical hyperbolicity. Theorem 1 ([HHP20]). Every HHG admits a proper cobounded action on an injective metric space. This has a number of interesting consequences for HHGs, and shows that many facts about hyperbolic groups remain true in this much broader setting. Corollary 2 ([HHP20]). Dehn’s conjugacy problem is soluble in HHGs. HHGs have only finitely many conjugacy classes of finite subgroups. A major theme in the geometric theory of groups in the 80s and 90s was the study of groups via distinguished classes of geodesics [Can84, GS91, Alo92, ECH`92]. One notable example is the theory of semihyperbolicity, as introduced by Alonso–Bridson in direct response to a call by Gromov for a weakened form of hyperbolicity [AB95]. A bicombing on a metric space pX, dq is a choice, for each x, y P X, of a uniform quasigeodesic γxy from x to y such that any two fellow-travel, in the sense that there is a constant K for which we have dpγxyptq, γzwptqq ď Kpdpx, zq ` dpy, wqq ` K for any γxy and γzw. A group is said to be semihyperbolic if it has a Cayley graph with an equivariant bicombing. Semihyperbolicity has several important consequences, and many groups are known to be semihyperbolic. However, the situation for mapping class groups remained unknown for 25 years, until the following, which is a consequence of Theorem 1 and work of Lang on bicombings in injective spaces. Corollary 3 ([HHP20]). HHGs are semihyperbolic. In particular, mapping class groups are semihyperbolic. For the natural subclass of colourable HHGs, which includes mapping class groups, semihyperbolicity was simultaneously and independently established by Durham–Minsky– Sisto, using very different methods [DMS20]. Theorem 1, and the methods used to establish it, open up some exciting possibilities for further study. The most optimistic of these would be to try to strengthen Corollary 3 to get biautomaticity, which is one of the strongest algorithmic properties that one can reasonably hope for a group to have. Its definition consists of asking for an “algorithmic” generating set (making the group automatic) that witnesses semihyperbolicity. It is a well-known theorem of Mosher that mapping class groups are automatic [Mos95], and 2 Research statement Harry Petyt semihyperbolicity is given by Corollary 3 or [DMS20], but the relevant generating sets are not known to agree. Question 4. Are mapping class groups biautomatic? Biautomaticity is known for groups acting geometrically on CAT(0) cube complexes [NR98], and for the class of Helly groups, which is related to that of groups acting properly coboundedly on injective spaces [CCG`20]. Conjecture 12, discussed below, may also be relevant for approaching Question 4. A second obvious line of investigation would be to ask about what other group-theoretic facts can be deduced from a proper cobounded action on an injective metric space. This is rather open-ended, but one particular topic I would like to investigate is the connection with recent work on (sublinearly) Morse geodesics and boundaries that generalises results of Charney–Sultan for CAT(0) spaces [CS15, Cor17, QRT20]. These concepts give a way to talk about the “hyperbolic directions” of a metric space. Problem 5. Understand the (sublinearly) Morse boundary of a cobounded injective space. For CAT(0) spaces, a more classical notion of boundary is the visual boundary [EO73, BH99], which is an important tool for understanding the large-scale geometry of such a space. In [DL15], Descombes–Lang constructed a version of the visual boundary for injective spaces with finite combinatorial dimension, which is a properness condition that permits the use of the Arzelà–Ascoli theorem. In view of Theorem 1, it would be interesting to try to weaken the assumption of finite combinatorial dimension to cover injective spaces that admit a proper cobounded group action. In particular, this would uncover new avenues for tackling questions about mapping class groups. Although the injective spaces in this setting cannot be assumed to be proper, the group action gives a quasiisometry to a proper space. Problem 6. Build a visual boundary for cobounded injective spaces. Question 6.1. Can you make use of the coarse properness coming from the group action? Question 6.2. What do you get in the case of mapping class groups? A third direction is suggested by the strategy of the proof of Theorem 1. By a powerful result of Behrstock–Hagen–Sisto [BHS21], HHGs admit a nice local approximation by CAT(0) cube complexes. Moreover, any CAT(0) cube complex equipped with the `8 metric is injective. The approach taken to Theorem 1 in [HHP20] is to change the metric on the given HHG in a way that is analogous to putting the `8 metric on the approximating cube complexes. Because p–normed spaces often have stronger properties than sup–normed spaces for p P p1, 8q (such as having strongly convex balls), it is natural to wonder about using the `p metric instead, and this is being investigated in ongoing work with Haettel and Hoda. A first step in this direction would be to answer the following. Question 7. Are CAT(0) cube complexes uniquely geodesic when given the `p–metric? A fourth line of inquiry is opened by the generality of the arguments of [HHP20], for many of them are made in the broader setting of coarse median spaces [Bow13] with a convex metric approximation by CAT(0) cube complexes. One of the most influential results of Masur–Minsky for mapping class groups is that the curve graph defined by Harvey [Har81] is hyperbolic [MM99], and studying the action of the mapping class group on the curve graph has proved to be extremely fruitful. An analogue of the curve graph has also been introduced for CAT(0) cube complexes [Hag14b, Gen20]. Working in the above more general setting, Spriano, Zalloum, and I are aiming to use some of the techniques of [HHP20] to produce a curve graph for such spaces. Question 8. Can you build a hyperbolic analogue of the curve graph for such a space X? Do Morse geodesics quasiisometrically embed in this graph? What extra assumptions are needed to deduce that X is hierarchically hyperbolic? 3 Research statement Harry Petyt

An important notion in the theory of hyperbolic groups is that of a quasiconvex sub- group, and there is a natural analogue in the setting of hierarchical hyperbolicity, namely a hierarchically quasiconvex subgroup. Since the seminal work of Sageev and others on producing actions of a group on CAT(0) cube complexes [Sag97, NR03, HW14], a key question to ask about subgroups of a group is whether they have bounded packing, which is a kind of coarse Helly property for left cosets. In the proof of Theorem 1 in [HHP20], the ball-intersection property needed for injectivity is deduced from a much more general result about hierarchical quasiconvexity, and this result allows us to conclude the following. Theorem 9 ([HHP20]). Hierarchically quasiconvex subgroups of HHGs have bounded packing. A recent result of Berlyne–Russell states that any graph product of HHGs is itself an HHG [BR20b]. More generally, they prove that every graph product of arbitrary groups is an HHG relative to the vertex groups. Relative hyperbolicity is a well-studied notion that was introduced by Gromov in his original essay on hyperbolic groups [Gro87]; the idea is that many non-hyperbolic groups (such as fundamental groups of cusped hyperbolic 3– manifolds) only fail to be hyperbolic because of some isolated regions on non-hyperbolicity (the incompressible tori). Relative hierarchical hyperbolicity is a natural generalisation. This suggests the following scheme, which would generalise results of [HW09]. Problem 10. Define a notion of relative hierarchical quasiconvexity and prove bounded packing for relatively hierarchically quasiconvex subgroups. This should apply to graphical subgroups of graph products.

3. Coarse CAT(0) cube complexes The class of hierarchically hyperbolic groups is in large part motivated by groups that act properly coboundedly on CAT(0) cube complexes (a.k.a. cubical groups). An interesting point is that the standard procedure for showing that a cubical group is an HHG always produces a hierarchy that entirely consists of quasitrees—graphs that are quasiisometric to trees. It is natural to ask whether this property actually characterises cubical HHGs: if the hierarchy of an HHG entirely consists of quasitrees, must the HHG be quasiisometric to a CAT(0) cube complex? In joint work with Hagen, we provide a positive answer to this question under a natural assumption. Theorem 11 ([HP19]). Any colourable HHG whose hierarchy consists entirely of qua- sitrees is quasiisometric to a finite-dimensional CAT(0) cube complex. The assumption of colourability is inspired by the cases of mapping class groups and vir- tually special groups, and all of the standard examples of HHGs are colourable [Hag21]. Us- ing similar methods in tandem with recent work of Bestvina–Bromberg–Fujiwara [BBF15, BBF19], I can show that Theorem 11 can be extended to cover mapping class groups. In fact, I am also currently working on a rather different approach that uses results on embed- dings of hyperbolic spaces in products of trees [BDS07] to remove the quasitree hypothesis from Theorem 11 and avoid the use of [BBF19]. Namely, I am confident that I can prove the following, which applies to mapping class groups. Conjecture 12. Every colourable HHG is quasiisometric to a finite-dimensional CAT(0) cube complex. This is already interesting just for mapping class groups, because a well-known result of Bridson states that they cannot act properly by semisimple isometries on complete CAT(0) spaces when the genus of the surface is at least three [Bri10]. However, conjugating the regular action by such a quasiisometry would give a proper cobounded quasiaction. The strategy that I plan to employ to prove Conjecture 12 should produce not just a quasiisometry, but a quasimedian quasiisometry. (Quasimedian maps are the natu- ral morphisms between coarse median spaces.) This stronger property should produce 4 Research statement Harry Petyt a correspondence between hierarchically quasiconvex subgroups of the HHG and convex subcomplexes of the cube complex. Using the Helly property for convex subcomplexes of CAT(0) cube complexes, this should lead to an alternative proof of bounded packing for hierarchically quasiconvex subgroups of colourable HHGs. On the subject of finding quasiisometries to CAT(0) cube complexes, I would like to try to use some of the ideas of [HP19] and my planned approach to Conjecture 12 to address the following question from Wise’s ICM talk of 2014. Question 13 ([Wis14]). Let G be a group that is hyperbolic relative to subgroups that are quasiisometric to CAT(0) cube complexes. Is G quasiisometric to a CAT(0) cube complex? It would also be interesting to consider the more general setting where G is a colourable HHG relative to subgroups that are quasiisometric to CAT(0) cube complexes, for example a graph product of such groups.

4. Structure of HHGs The success of hierarchical hyperbolicity as a methodology for studying groups of interest means that it is worth trying to understand the machinery itself better. A hierarchically hyperbolic structure is the collection of data associated with an HHG, minus the data of the group action. With Spriano, we addressed the question of what possible hierarchically hyperbolic structures can arise for an HHG, and proved a decomposition theorem similar to that for CAT(0) cube complexes [CS11]. Theorem 14 ([PS20]). If G is an HHG, then every HHG structure on G coarsely decom- poses as a product of “irreducible” hierarchically hyperbolic structures. This greatly clarifies both the structure of a general HHG and the possible HHG struc- tures on a given group. It is also noteworthy that the proof given in [PS20] uses only elementary tools for HHGs, so Theorem 14 can be viewed as foundational. One way in which structural results are useful in any theory is that they put constraints on which objects can satisfy them, and Theorem 14 is no exception. It is fairly straight- forward to use the new obstructions to hierarchical hyperbolicity provided by Theorem 14 to give a new proof of the following that does not use Tits’ alternative [?]. Corollary 15 ([PS20]). Infinite torsion groups are not HHGs. A classical family of groups are the crystallographic groups, which are defined to be discrete subgroups of Isom Rn that act properly cocompactly on Rn. By considering HHGs that have abelian subgroups at finite index, Theorem 14 can be used together with a result of Hagen [Hag14a] to characterise exactly which crystallographic groups are HHGs. Theorem 16 ([PS20]). A crystallographic group is hierarchically hyperbolic if and only if its point group embeds in OnpZq. This characterisation yields the first examples of groups that are not HHGs but have finite-index subgroups that are HHGs, such as the p3, 3, 3q–triangle Coxeter group. Since Theorem 16 first appeared, work of Hoda [Hod20] has made it possible to deduce Theo- rem 16 from Theorem 1. In another vein, Theorem 14 also provides a new approach to proving a hierarchical version of the long-standing rank-rigidity conjecture of Ballmann–Buyalo for CAT(0) spaces [BB08]. Although such a result was previously established in [DHS17], the arguments there are significantly more involved and less self-contained than those of [PS20], which use only fundamental results about HHGs. The theorem can be formulated as follows. Theorem 17 ([PS20]). If G is an HHG that is not virtually cyclic, then G is either acylindrically hyperbolic or wide. 5 Research statement Harry Petyt

Being wide is a natural coarse generalisation of being a direct product of infinite groups. Acylindrical hyperbolicity is a weak form of negative curvature that covers a large range of examples [Osi16]. The main open question of the theory of acylindrical hyperbolicity is whether it is preserved by quasiisometries. In the restricted setting of HHGs, this is confirmed by Theorem 17. Corollary 18 ([PS20]). If G and H are quasiisometric HHGs, then G is acylindrically hyperbolic if and only if H is. The improved perspective afforded by Theorem 14 has already been productive for the study of growth of groups. The growth of a group describes how quickly the balls in its Cayley graphs grow as the radius increases; for finitely generated groups this is bounded above by the growth rate of a free group, which is exponential. However, the exact growth rate depends on the choice of generating set. If a group has an exponential lower bound on its growth that is independent of the choice of generating set, then it is said to have uniform exponential growth. Uniform exponential growth is well understood in many classes of groups, including hyperbolic groups [Kou98], linear groups [EMO05], and mapping class groups [AAS07]. In [KS19], Kar–Sageev raised the question of which groups that act on CAT(0) cube complexes have uniform exponential growth, and answered the question for groups act- ing freely on 2–dimensional CAT(0) cube complexes. In the setting of cubical groups, a stronger, algebraic version of Theorem 14 is known to hold [NS13, CM09]. In joint work with Gupta, by specialising partial results for HHGs [ANS19] to the case of cubical groups that are HHGs, we used this perspective to prove the following. Theorem 19 ([GP21]). Let G be a group acting freely cocompactly on a locally finite, finite- dimensional CAT(0) cube complex X, and assume that X has a factor system. Either G is virtually abelian, or G has uniform exponential growth. The assumption that X has a factor system ensures that G is an HHG. It is very weak, and is satisfied by all known cocompact cube complexes [HS20], but the question of whether it is always satisfied turns out to be surprisingly delicate. Question 20. Can similar arguments be applied to all HHGs? As a stepping stone, one may wish to prove a technical result of the following form. Subquestion. Suppose that G is an HHG that is quasiisometric to X ˆ Z for some space X with a hierarchically hyperbolic structure. Up to finite-index subgroups and finite kernels, is G a central extension of an HHG by Z? In general, it is much more difficult to understand the subgroups of a group than the group itself, as their geometry may not be faithfully represented by that of the ambient group. In spite of this, with Spriano we were able to prove an analogue of Theorem 14 for subgroups of HHGs [PS20]. It is known from work of Mangahas that finitely generated subgroups of mapping class groups have uniform exponential growth whenever they are not virtually abelian [Man10]. This is known as uniform uniform exponential growth. It would be interesting to investigate whether the results with Spriano can be used to generalise this to other HHGs. Question 21. When do HHGs have uniform uniform exponential growth?

Items coauthored by Harry Petyt [GP21] Radhika Gupta and Harry Petyt. Uniform exponential growth for cocompactly cubu- lated groups. Appended to “Hierarchically hyperbolic groups and uniform exponential growth” by Carolyn Abbott, Thomas Ng, and Davide Spriano. Preprint available at people.maths.bris.ac.uk/~aj18755/UEG.pdf, 2021. 6 Research statement Harry Petyt

[HHP20] Thomas Haettel, Nima Hoda, and Harry Petyt. The coarse Helly property, hierarchical hyper- bolicity, and semihyperbolicity. arXiv preprint arXiv:2009.14053, 2020. [HP19] Mark Hagen and Harry Petyt. Projection complexes and quasimedian maps. arXiv:2108.13232. To appear in Algebr. Geom. Topol., 2019. [PS20] Harry Petyt and Davide Spriano. Unbounded domains in hierarchically hyperbolic groups. arXiv preprint arXiv:2007.12535, 2020.

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