Universitat Wien

Christopher H. Cashen

picture of Christopher Cashen
Christopher Cashen, PhD
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
Email: christopher DOT cashen AT univie.ac.at
Office: 2.130

I am a Senior Research Fellow in the Geometric and Analytic Group Theory group at the University of Vienna. I help organize our Seminar.

Curriculum Vitae


Page numbers link to the final published version on the publisher's webpage. Open access logos (Open Access logo) link to open access versions, as permitted. You can also find my papers on the arXiv.
Quasi-isometries Between Tubular Groups
Groups, Geometry and Dynamics, 4 (2010), no. 3, 473–516. Open Access logo
(Abstract)We give a quasi-isometry classification of graphs of virtual \(\mathbb{Z}^2\)'s amalgamated over virtually cyclic subgroups.
Line Patterns in Free Groups, w/ Nataša Macura
Geometry & Topology, 15 (2011), no. 3, 1419–1475. Open Access logo
(Abstract)Given an algebraic lamination \(\mathcal{L}\) of the free group \(\mathbb{F}\), exactly one of the following is true:
  • \(\mathbb{F}\) splits freely or cyclically relative to \(\mathcal{L}\).
  • \(\mathbb{F}\) is quasi-isometrically rigid relative to \(\mathcal{L}\).
  • \(\mathbb{F}\) is the fundamental group of a thrice-punctured sphere and leaves of \(\mathcal{L}\) are the peripheral subgroups.
Virtual Geometricity is Rare, w/ Jason F. Manning
LMS Journal of Computation and Mathematics, 18 (2015), no. 1, 444–455. Open Access logo
(Abstract) An element in the free group \(\mathbb{F}\) is geometric if it can be represented by an embedded curve in the boundary of a handlebody with fundamental group \(\mathbb{F}\). It is virtually geometric if it becomes geometric upon lifting to some finite index subgroup of \(\mathbb{F}\). The probability that a random element is virtually geometric decays to zero exponentially quickly with respect to the word length of the element with respect to a fixed basis. We also give experimental estimates for the rate of decay; these are obtained by our computer program that checks virtual geometricity for a given input word.
Growth Tight Actions, w/ Goulnara N. Arzhantseva and Jing Tao
Pacific Journal of Mathematics, 278 (2015), no. 1, 1–49. Open Access logo
(Abstract)A proper action of a group \(G\) on a metric space \(X\) is growth tight if for every infinite index normal subgroup \(N\) of \(G\) the exponential growth rate of \(G\) with respect to \(X\), ie, of an orbit of \(G\) in \(X\), is strictly greater than the corresponding growth rate of \(G/N\). We show that an action is growth tight if it has a strongly contracting element and if there is an orbit that is sufficiently convex. The latter condition is satisfied if there is a quasi-convex orbit, but also in other interesting cases such as the action of the mapping class group on Teichmüller space.
Mapping Tori of Free Group Automorphisms, and the Bieri-Neumann-Strebel Invariant of Graphs of Groups, w/ Gilbert Levitt
Journal of Group Theory, 19 (2016), no. 2, 191–216. Open Access logo
(Abstract)We compute Bieri-Neumann-Strebel (BNS) invariants for certain graphs of groups and hierarchies of groups. In particular, we show that the BNS invariant of the mapping torus of a polynomially growing free group automorphism is the complement of finitely many rationally defined subspheres.
Splitting Line Patterns in Free Groups
Algebraic & Geometric Topology, 16 (2016), no. 2, 621–673. Open Access logo
(Abstract)Techniques of Paper 2 are used to derive a canonical JSJ decomposition of a free group \(\mathbb{F}\) relative to an algebraic lamination \(\mathcal{L}\) in terms of the topology of the quotient \(\partial\mathbb{F}/\partial\mathcal{L}\). As an application, we characterize virtual geometricity (cf Paper 3) as having a relative JSJ decomposition with geometric pieces.
Growth Tight Actions of Product Groups, w/ Jing Tao
Groups, Geometry and Dynamics, 10 (2016), no. 2, 753–770. Open Access logo
(Abstract)If \(G=\prod_{i=1}^nG_i\) is a product of finitely generated groups and for each \(i\) we have a proper, cocompact action of \(G_i\) on a metric space \(X_i\) with a strongly contracting element then the product action of \(G\) on \(X=\prod_{i=1}^nX_i\) is growth tight if \(X\) is given the \(L^p\) metric for \(p>1\) and not growth tight if \(X\) is given the \(L^1\) metric. In particular, if the \(X_i\) are Cayley graphs of the \(G_i\) then the \(L^1\) metric and the \(L^\infty\) metric both correspond to word metrics on \(G\). This provides the first construction of a group such that the action on one of its Cayley graphs is growth tight and the action on another of its Cayley graphs is not.
Quasi-isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology
Analysis and Geometry in Metric Spaces, 4 (2016), no. 1, 278–281. Open Access logo
(Abstract) We consider a 'contracting boundary' of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space. We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.
Quasi-isometries Between Groups with Two-Ended Splittings, w/ Alexandre Martin
Mathematical Proceedings of the Cambridge Philosophical Society, 162 (2017), no. 2, 249–291. Open Access logo
(Abstract)We construct invariants for boundary homeomorphism and quasi-isometry of hyperbolic groups that split over two-ended subgroups in terms of the respective homeomorphism/quasi-isometry types of the vertex groups relative to the edge groups. For boundary homeomorphism we get a complete invariant. For quasi-isometry we get a complete invariant when the vertex groups are rigid relative to their incident edge groups.
A Geometric Proof of the Structure Theorem for Cyclic Splittings of Free Groups
Topology Proceedings, 50 (2017), 335–349. Open Access logo
(Abstract)Application of the techniques of Papers 2 and 6 giving an alternate proof that an amalgam of two free groups over a cyclic subgroup is free if and only if the edge group is a free factor of one of the vertex groups. Applies also to virtually free groups amalgamated over a virtually cyclic subgroup.
Characterizations of Morse Quasi-geodesics via Superlinear Divergence and Sublinear Contraction, w/ Goulnara N. Arzhantseva, Dominik Gruber, and David Hume
Documenta Mathematica, 22 (2017), 1193–1224. Open Access logo
(Abstract)We initiate a systematic study of contracting projections. We show that sublinear contraction is equivalent to the well-known Morse property and to the property of having completely superlinear divergence. We prove sublinear analogues of several theorem about strongly contracting geodesics.
Quasi-isometry Classification for [Right-Angled Coxeter Groups defined by suitable subdivisions of] Complete Graphs, w/ Pallavi Dani and Anne Thomas, appendix to Bowditch's JSJ Tree and the Quasi-isometry Classification of Certain Coxeter Groups by Dani and Thomas
Journal of Topology (in press). preprint
(Abstract) In the main paper Dani and Thomas show how to derive the JSJ decomposition of a hyperbolic 2–dimensional right-angled Coxeter group from its defining graph. In the appendix we give a complete quasi-isometry classification of hyperbolic 2–dimensional right-angled Coxeter groups that are defined by subdivisions of complete graphs.
A Metrizable Topology on the Contracting Boundary of a Group, w/ John M. Mackay
Transactions of the American Mathematical Society (in press). Open Access logo
(Abstract) The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.
Negative Curvature in Graphical Small Cancellation Groups, w/ Goulnara N. Arzhantseva, Dominik Gruber, and David Hume
submitted. preprint
(Abstract) We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical \(Gr'(1/6)\) small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically.

We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group \(G\) containing an element that is strongly contracting with respect to one finite generating set of \(G\) and not strongly contracting with respect to another. In the case of classical \(C'(1/6)\) small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting.

We show that many graphical \(Gr'(1/6)\) small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups.

In the course of our analysis we show that if the defining graph of a graphical \(Gr'(1/6)\) small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.

Cogrowth for Group Actions with Strongly Contracting Elements, w/ Goulnara N. Arzhantseva
submitted. preprint
(Abstract) We show that for a finitely generated group \(G\) acting properly on a geodesic metric space \(X\) with a strongly contracting element and purely exponential growth, and for every infinite normal subgroup \(N\) of \(G\), the growth rates \(\delta_G\) and \(\delta_N\) of the orbits of \(G\) and \(N\) in \(X\) satisfy \(\delta_N/\delta_G > 1/2\). This generalizes several results where the same conclusion is obtained for \(G\) a free group or \(X\) a negatively curved space.

Notes on the arXiv

Computing the Maximum Slope Invariant in Tubular Groups (2009).
(Abstract)Addendum to Paper 1. Gives an example of two tubular groups that are distinguished by the ‘maximum slope invariant’ introduced there but not by Dehn function or other obvious quasi-isometry invariants.


virtuallygeometric, w/ Jason F. Manning, Computer Program, (2014).
(Description) This code is for testing whether a multiword in a free group is virtually geometric. Towards this end it implements various algorithms related to elements and subgroups of finitely generated free groups, including computing the JSJ decomposition of a free group relative to a multiword.

Other Stuff

Slides from 'Line Patterns in Free Groups' talk, Joint Meetings, Jan 2011.
Video of 'The topology of the contracting boundary of a group' talk from conference at Isaac Newton Institute, Jan 2017.


I am not currently teaching. Visit my teaching page from Utah instead.


I gratefully acknowledge support from:
Logo of
   the Austrian Science Fund (FWF) (2017-2020) Austrian Science Fund (FWF): P 30487-N35
(2014-2016) Austrian Science Fund (FWF): M 1717-N25, Lise Meitner Fellowship.
(2012-2014) European Research Council (ERC) grant of Goulnara ARZHANTSEVA, "ANALYTIC" grant agreement n259527.
(2011-2012) French National Research Agency (ANR) grant: ANR-2010-BLAN-116-01 GGAA
(2008-2011) USA National Science Foundation (NSF) VIGRE grant to the University of Utah.
This page created and maintained by christopher DOT cashen AT univie.ac.at
Last updated  June 30, 2017.