Research statement Harry Petyt 1. Research setting My research is in the area of geometric group 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 metric space 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 Cayley graph 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 hyperbolic group. 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 group action 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 ¤ 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.