Research

Book/conference proceedings

Beyond Hyperbolicity. Co-edited with Richard Webb and Henry Wilton.

Recent papers

A remark on thickness of free-by-cyclic groups. Illinois J. Math. 63:4, 633-643, 2019. (journal link)

Panel collapse and its applications. With Nicholas Touikan. Groups, Geometry, and Dynamics. 13:4, 1285-1334, 2019. (journal link)

On hierarchical hyperbolicity of cubical groups. With Tim Susse. Isr. J. Math. 236, 45-89, 2020. (journal link)
This paper characterises cocompact cubulations with factor systems. More recently, Sam Shepherd has produced the first examples without factor systems, using a phenomenon present in some CSCs.
Quasiflats in hierarchically hyperbolic spaces. With Jason Behrstock and Alessandro Sisto. Duke Math. Jour. 170:5, 909-996, 2021. (journal link)
The "cubical approximation theorem" proved in this paper was re-proved here by Brian Bowditch, who has also extended the result on quasiflats to the setting of coarse median spaces. Additional related developments explained by Durham.
Dehn filling Dehn twists. With François Dahmani and Alessandro Sisto. Proc. A. Royal Soc. Edinburgh. 151:1, 28-51, 2021. (journal link)

Deforming cubulations of hyperbolic groups. With Elia Fioravanti. J. Topol. 14:3, 877-912, 2021. (journal link)

Large facing tuples and a strengthened sector lemma. Tunisian Jour. Math. 4:1, 55-86, 2022. (journal link)

Acylindrical hyperbolicity of cubical small-cancellation groups. With Goulnara Arzhantseva. Alg. Geom. Topol. 22:5, 2007-2078, 2022. (journal link)

Projection complexes and quasimedian maps. With Harry Petyt. Alg. Geom. Topol. 22:7, 3277-3304, 2022. (journal link)
Petyt significantly extends these ideas here.
Extra-large type Artin groups are hierarchically hyperbolic. With Alexandre Martin and Alessandro Sisto. Math. Ann. 388, 867-938, 2024. (journal link)

Non-colourable hierarchically hyperbolic groups. Internat. Jour. Alg. Comp. 33:02, 337-350, 2023. (journal link)

Some examples of separable convex-cocompact subgroups. With Alessandro Sisto. Bull. Lond. Math. Soc. 55:5, 2242-2257, 2023. (journal link)

Homotopy equivalent boundaries of cube complexes. With Talia Fernós and David Futer. Geom. Ded. 218:33, 2024. (journal link)

Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms. With Jacob Russell, Alessandro Sisto and Davide Spriano. Ann. de l'inst. Fourier. To appear, 2023.

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups. With Jason Behrstock, Alexandre Martin, Alessandro Sisto. Preprint, 2020.

Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups. With Carolyn Abbott, Harry Petyt, and Abdul Zalloum. Preprint, 2023.

A combinatorial structure for many hierarchically hyperbolic spaces. With Giorgio Mangioni and Alessandro Sisto. Preprint, 2023.
This paper is about partial converses to the main theorem in the above paper about combinatorial HHSes. Interestingly, one question left hanging can be phrased purely in terms of orthomodular lattices; this question is at the very end and I would like to advertise it.
Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity. With Antoine Goldsborough, Harry Petyt and Alessandro Sisto. Appendix by Jacob Russell. Preprint, 2023.

Real cubings and asymptotic cones of hierarchically hyperbolic groups. With Montserrat Casals-Ruiz and Ilya Kazachkov. Working draft, 2022.

Older papers

Special groups with an elementary hierarchy are virtually free-by-Z. With Dani Wise. Groups, Geometry, and Dynamics 4:3, 597-603, 2010.

Weak hyperbolicity of cube complexes and quasi-arboreal groups. J. Topol. 7:2, 385-418, 2014.

On embedding CAT(0) cube complexes into products of trees via colouring their hyperplanes. With Victor Chepoi. J. Comb. Theor. Ser. B 103:4, 428-467, 2013.
Summary in Proceedings of the Third Conference of Mathematical Society of Moldova. Notes, by Chepoi, based on the above paper and this related paper of his.
The simplicial boundary of a CAT(0) cube complex. Alg. Geom. Topol. 13:3, 1299-1367, 2013.
Corrigendum on Theorem 3.10 in the infinite-dimensional case; finite-dimensional case is unaffected (journal version).
Cocompactly cubulated crystallographic groups. J. Lond. Math. Soc. 90 (1): 140-166, 2014.
Result recently strengthened in this paper by Nima Hoda.
Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries. With Jason Behrstock. Groups, Geometry, and Dynamics. (10:2) 649-707, 2016.
Terminology clarification: the word "quasiconvex" is used in two ways in the paper, sometimes confusingly. In arguments not about group actions, it means "any two points in the subspace are joined by a quasigeodesic in the subspace". For a subgroup H of a group G acting geometrically on a cube complex X, "quasiconvex" is intended in the stronger, cubical sense: H-orbits are Hausdorff close to their convex hull in X. Nowadays, one would say "cubically convex-cocompact".
Cocompactly cubulated graph manifolds. With Piotr Przytycki. Isr. J. Math. 207 (1): 377-394, 2015.

Erratum correcting the proof of Lemma 4.7(2) (journal version).
Cubulating hyperbolic free-by-cyclic groups: the irreducible case. With Dani Wise. Duke Math. Jour. (165:9) 1753-1813, 2016.

Thickness, relative hyperbolicity, and randomness in Coxeter groups. With Jason Behrstock and Alessandro Sisto, appendix with Pierre-Emmanuel Caprace. Alg. Geom. Topol. (17:2) 705-740, 2016.

Software associated with the above paper.
Residual finiteness growths of virtually special groups. With Khalid Bou-Rabee and Priyam Patel. Math. Zeit. 279 (1-2): 297-310, 2015.

Cubulating hyperbolic free-by-cyclic groups: the general case. With Dani Wise. Geom. Funct. Anal. 25:1, 134-179, 2015.

The main result now follows from a more general result due to François Dahmani, Suraj Krishna M S, and Jean Pierre Mutanguha. In the free-by-cyclic case, their proof is simpler than ours, because it uses the irreducible case as a base case, and then uses relative cubulation technology to avoid the relative train track machinery. And hence it also applies more generally.
Hierarchically hyperbolic spaces I: curve complexes for cubical groups. With Jason Behrstock and Alessandro Sisto. Geom. Topol. (21:3) 1731-1804, 2017.

Quantifying separability in virtually special groups. With Priyam Patel. Pacific Jour. Math. (284:1) 103-120, 2016.

This is a slightly different proof of Theorem A. The main idea is the same, but the argument is modified in a way that avoids some technicalities. Not for publication, just a record for possible later use.
Global structural properties of random graphs. With Jason Behrstock, Victor Falgas-Ravry, and Tim Susse. Int. Math. Res. Not. 2018:5, 1411-1441, 2018.

Source code and data alluded to in the above paper.
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula. With Jason Behrstock and Alessandro Sisto. Pacific J. Math. (299:2) 257-338, 2019.

Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups. With Jason Behrstock and Alessandro Sisto. Proc. Lond. Math. Soc. (114:5) 890-926, 2017.

Boundaries and automorphisms of hierarchically hyperbolic spaces. With Matthew Durham and Alessandro Sisto. Geom. Topol. (21:6) 3659-3758, 2017.

Corrigendum fixing an error affecting the proofs of Theorem 7.1 and Theorem 9.15 (journal version).

Some point-set topology comments, mainly explaining why the boundary of a proper HHS is metrisable. Not for publication since it's just giving details of ideas in the paper, but possibly helpful to someone reading Sections 2 and 3.
Cubulating mapping tori of polynomial growth free group automorphisms. With Dani Wise. Preprint, 2018.

This preprint is under repair (inductive step as written doesn't work). Will update here and on ArXiV when repairs are complete.

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	Book/conference proceedings Beyond Hyperbolicity. Co-edited with Richard Webb and Henry Wilton.
	Recent papers A remark on thickness of free-by-cyclic groups. Illinois J. Math. 63:4, 633-643, 2019. (journal link) Panel collapse and its applications. With Nicholas Touikan. Groups, Geometry, and Dynamics. 13:4, 1285-1334, 2019. (journal link) On hierarchical hyperbolicity of cubical groups. With Tim Susse. Isr. J. Math. 236, 45-89, 2020. (journal link) This paper characterises cocompact cubulations with factor systems. More recently, Sam Shepherd has produced the first examples without factor systems, using a phenomenon present in some CSCs. Quasiflats in hierarchically hyperbolic spaces. With Jason Behrstock and Alessandro Sisto. Duke Math. Jour. 170:5, 909-996, 2021. (journal link) The "cubical approximation theorem" proved in this paper was re-proved here by Brian Bowditch, who has also extended the result on quasiflats to the setting of coarse median spaces. Additional related developments explained by Durham. Dehn filling Dehn twists. With François Dahmani and Alessandro Sisto. Proc. A. Royal Soc. Edinburgh. 151:1, 28-51, 2021. (journal link) Deforming cubulations of hyperbolic groups. With Elia Fioravanti. J. Topol. 14:3, 877-912, 2021. (journal link) Large facing tuples and a strengthened sector lemma. Tunisian Jour. Math. 4:1, 55-86, 2022. (journal link) Acylindrical hyperbolicity of cubical small-cancellation groups. With Goulnara Arzhantseva. Alg. Geom. Topol. 22:5, 2007-2078, 2022. (journal link) Projection complexes and quasimedian maps. With Harry Petyt. Alg. Geom. Topol. 22:7, 3277-3304, 2022. (journal link) Petyt significantly extends these ideas here. Extra-large type Artin groups are hierarchically hyperbolic. With Alexandre Martin and Alessandro Sisto. Math. Ann. 388, 867-938, 2024. (journal link) Non-colourable hierarchically hyperbolic groups. Internat. Jour. Alg. Comp. 33:02, 337-350, 2023. (journal link) Some examples of separable convex-cocompact subgroups. With Alessandro Sisto. Bull. Lond. Math. Soc. 55:5, 2242-2257, 2023. (journal link) Homotopy equivalent boundaries of cube complexes. With Talia Fernós and David Futer. Geom. Ded. 218:33, 2024. (journal link) Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms. With Jacob Russell, Alessandro Sisto and Davide Spriano. Ann. de l'inst. Fourier. To appear, 2023. A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups. With Jason Behrstock, Alexandre Martin, Alessandro Sisto. Preprint, 2020. Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups. With Carolyn Abbott, Harry Petyt, and Abdul Zalloum. Preprint, 2023. A combinatorial structure for many hierarchically hyperbolic spaces. With Giorgio Mangioni and Alessandro Sisto. Preprint, 2023. This paper is about partial converses to the main theorem in the above paper about combinatorial HHSes. Interestingly, one question left hanging can be phrased purely in terms of orthomodular lattices; this question is at the very end and I would like to advertise it. Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity. With Antoine Goldsborough, Harry Petyt and Alessandro Sisto. Appendix by Jacob Russell. Preprint, 2023. Real cubings and asymptotic cones of hierarchically hyperbolic groups. With Montserrat Casals-Ruiz and Ilya Kazachkov. Working draft, 2022.
	Older papers Special groups with an elementary hierarchy are virtually free-by-Z. With Dani Wise. Groups, Geometry, and Dynamics 4:3, 597-603, 2010. Weak hyperbolicity of cube complexes and quasi-arboreal groups. J. Topol. 7:2, 385-418, 2014. On embedding CAT(0) cube complexes into products of trees via colouring their hyperplanes. With Victor Chepoi. J. Comb. Theor. Ser. B 103:4, 428-467, 2013. Summary in Proceedings of the Third Conference of Mathematical Society of Moldova. Notes, by Chepoi, based on the above paper and this related paper of his. The simplicial boundary of a CAT(0) cube complex. Alg. Geom. Topol. 13:3, 1299-1367, 2013. Corrigendum on Theorem 3.10 in the infinite-dimensional case; finite-dimensional case is unaffected (journal version). Cocompactly cubulated crystallographic groups. J. Lond. Math. Soc. 90 (1): 140-166, 2014. Result recently strengthened in this paper by Nima Hoda. Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries. With Jason Behrstock. Groups, Geometry, and Dynamics. (10:2) 649-707, 2016. Terminology clarification: the word "quasiconvex" is used in two ways in the paper, sometimes confusingly. In arguments not about group actions, it means "any two points in the subspace are joined by a quasigeodesic in the subspace". For a subgroup H of a group G acting geometrically on a cube complex X, "quasiconvex" is intended in the stronger, cubical sense: H-orbits are Hausdorff close to their convex hull in X. Nowadays, one would say "cubically convex-cocompact". Cocompactly cubulated graph manifolds. With Piotr Przytycki. Isr. J. Math. 207 (1): 377-394, 2015. Erratum correcting the proof of Lemma 4.7(2) (journal version). Cubulating hyperbolic free-by-cyclic groups: the irreducible case. With Dani Wise. Duke Math. Jour. (165:9) 1753-1813, 2016. Thickness, relative hyperbolicity, and randomness in Coxeter groups. With Jason Behrstock and Alessandro Sisto, appendix with Pierre-Emmanuel Caprace. Alg. Geom. Topol. (17:2) 705-740, 2016. Software associated with the above paper. Residual finiteness growths of virtually special groups. With Khalid Bou-Rabee and Priyam Patel. Math. Zeit. 279 (1-2): 297-310, 2015. Cubulating hyperbolic free-by-cyclic groups: the general case. With Dani Wise. Geom. Funct. Anal. 25:1, 134-179, 2015. The main result now follows from a more general result due to François Dahmani, Suraj Krishna M S, and Jean Pierre Mutanguha. In the free-by-cyclic case, their proof is simpler than ours, because it uses the irreducible case as a base case, and then uses relative cubulation technology to avoid the relative train track machinery. And hence it also applies more generally. Hierarchically hyperbolic spaces I: curve complexes for cubical groups. With Jason Behrstock and Alessandro Sisto. Geom. Topol. (21:3) 1731-1804, 2017. Quantifying separability in virtually special groups. With Priyam Patel. Pacific Jour. Math. (284:1) 103-120, 2016. This is a slightly different proof of Theorem A. The main idea is the same, but the argument is modified in a way that avoids some technicalities. Not for publication, just a record for possible later use. Global structural properties of random graphs. With Jason Behrstock, Victor Falgas-Ravry, and Tim Susse. Int. Math. Res. Not. 2018:5, 1411-1441, 2018. Source code and data alluded to in the above paper. Hierarchically hyperbolic spaces II: Combination theorems and the distance formula. With Jason Behrstock and Alessandro Sisto. Pacific J. Math. (299:2) 257-338, 2019. Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups. With Jason Behrstock and Alessandro Sisto. Proc. Lond. Math. Soc. (114:5) 890-926, 2017. Boundaries and automorphisms of hierarchically hyperbolic spaces. With Matthew Durham and Alessandro Sisto. Geom. Topol. (21:6) 3659-3758, 2017. Corrigendum fixing an error affecting the proofs of Theorem 7.1 and Theorem 9.15 (journal version). Some point-set topology comments, mainly explaining why the boundary of a proper HHS is metrisable. Not for publication since it's just giving details of ideas in the paper, but possibly helpful to someone reading Sections 2 and 3. Cubulating mapping tori of polynomial growth free group automorphisms. With Dani Wise. Preprint, 2018. This preprint is under repair (inductive step as written doesn't work). Will update here and on ArXiV when repairs are complete.