# Matthew Cordes

I am a fellow of the ETH Zürich Postdoctoral Fellowship Program under the mentorship of Alessandra Iozzi.

Before I was at ETHZ, I was a Zuckerman STEM Leadership Postdoctoral Scholar at the Technion Department of Mathematics. I completed my Ph.D. at Brandeis University under the supervision of Ruth Charney.

You can find my CV here.

## Research

I work in the field of geometric group theory. My research interests include groups and spaces with some aspect of negative curvature and their boundaries.

### Papers

#### Foundations of approximate geometric group theory

Joint with Tobias Hartnick, Vera Tonić, in preparation

#### Cannon–Thurston maps for CAT(0) groups with isolated flats

Joint with Benjamin Beeker, Giles Gardam, Radhika Gupta, Emily Stark

Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon–Thurston maps do not exist for infinite torsion-free normal hyperbolic subgroups of a non-hyperbolic CAT(0) group with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston maps do not exist for infinite torsion-free infinite-index normal CAT(0) subgroups with isolated flats in a non-hyperbolic CAT(0) group with isolated flats. We determine the isomorphism types of the normal subgroups in the above two settings.

#### Quasi-mobius homeomorphisms of Morse boundaries

To appear Bulletin of the London Mathematical Society

The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic-like behavior in the space. A key property of this boundary is that a quasi-isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper we investigate when the converse holds. We prove that for $$X,Y$$ proper, cocompact spaces, a homeomorphism between their Morse boundaries is induced by a quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.

#### A survey on Morse boundaries & stability

To appear "Beyond Hyperbolicity" LMS Lecture Notes

In this paper we survey many of the known results about Morse boundaries and stability.

#### Relatively hyperbolic groups with fixed peripherals

Joint with David Hume
Israel Journal of Mathematics

We build quasi-isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors.

We prove that, given any finite collection of finitely generated groups $$\mathcal{H}$$ each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi-isometry types of one-ended groups which are hyperbolic relative to $$\mathcal{H}$$.

#### Boundary convex cocompactness and stability of subgroups of finitely generated groups

International Mathematics Research Notices

A Kleinian group $$\Gamma < \mathrm{Isom}(\mathbb H^3)$$ is called convex cocompact if any orbit of $$\Gamma$$ in $$\mathbb H^3$$ is quasiconvex or, equivalently, $$\Gamma$$ acts cocompactly on the convex hull of its limit set in $$\partial \mathbb H^3$$.

Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups.

Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.

#### Stability and the Morse boundary

Joint with David Hume
Journal of the London Mathematical Society

Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalize these strategies by viewing any geodesic metric space as a countable union of stable subspaces: we show that every stable subgroup is a quasi-convex subset of a set in this collection and that the Morse boundary is recovered as the direct limit of the usual Gromov boundaries of these hyperbolic subspaces.

We use this approach, together with results of Leininger–Schleimer, to deduce that there is no purely geometric obstruction to the existence of a non-virtually-free convex cocompact subgroup of a mapping class group.

In addition, we define two new quasi-isometry invariant notions of dimension: the stable dimension, which measures the maximal asymptotic dimension of a stable subset; and the Morse capacity dimension, which naturally generalises Buyalo's capacity dimension for boundaries of hyperbolic spaces.

We prove that every stable subset of a right-angled Artin group is quasi-isometric to a tree; and that the stable dimension of a mapping class group is bounded from above by a multiple of the complexity of the surface. In the case of relatively hyperbolic groups we show that finite stable dimension is inherited from peripheral subgroups.

Finally, we show that all classical small cancellation groups and certain Gromov monster groups have stable dimension at most 2.

#### Morse boundaries of proper geodesic spaces

Groups, Geometry, and Dynamics

We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper $$\mathrm{CAT}(0)$$ space this boundary is the contracting boundary of Charney and Sultan, and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichmüller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichmüller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichmüller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichmüller space into the Thurston compactification of Teichmüller space by projective measured foliations.

#### Random nilpotent groups I

Joint with Moon Duchin, Yen Duong, Meng-Che Ho, Ayla P. Sánchez
International Mathematics Research Notices

We study random nilpotent groups in the well-established style of random groups, by choosing relators uniformly among freely reduced words of (nearly) equal length and letting the length tend to infinity. Whereas random groups are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients of a free nilpotent group. This model reveals new phenomena because nilpotent groups are not "visible" in the standard model of random groups (due to the sharp phase transition from infinite hyperbolic to trivial groups).

## Teaching

• Spring 2019: Fuchsian Groups (ETHZ)
• Fall 2014: Calculus I (Brandeis University)
• Fall 2013: Precalculus (Brandeis University)
• Spring 2013: Calculus II (Brandeis University)
• Fall 2012: Calculus II (Brandeis University)