# Christopher H. Cashen

Fakultät für Mathematik
Universität Wien
Oskar-Morgenstern-Platz 1
1090 Wien, Österreich
Email: christopher DOT cashen AT univie.ac.at
Office: 2.130

I am a postdoc in the Geometric and Analytic Group Theory group at the University of Vienna.

### Papers (show descriptions)

Papers posted here are author versions. Final published versions may differ.
Quasi-isometries Between Tubular Groups, Groups, Geometry and Dynamics 4 (2010) no. 3, 473–516. doi:10.4171/GGD/92
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. doi:10.2140/gt.2011.15.1419
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. doi:10.1112/S1461157015000108
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. doi:10.2140/pjm.2015.278.1
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, 26pp, in press. doi:10.1515/jgth-2015-0038
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.
Growth Tight Actions of Product Groups (w/ Jing Tao), Groups, Geometry and Dynamics, 15pp, in press.
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.
Splitting Line Patterns in Free Groups, Algebraic & Geometric Topology, 56pp, in press.
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.
Characterizations of Morse Quasi-geodesics via Superlinear Divergence and Sublinear Contraction (w/ Goulnara N. Arzhantseva, Dominik Gruber, and David Hume), submitted, 24pp.
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-isometries Between Groups with Two-Ended Splittings (w/ Alexandre Martin), preprint, 61pp.
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.
Contracting Geodesics in Graphical Small Cancellation Groups (w/ Goulnara N. Arzhantseva, Dominik Gruber, and David Hume), in preparation.
We give necessary and sufficient conditions for a geodesic in a graphical small cancellation group to be contracting. Some consequences are:
• Many $$Gr'(1/6)$$ graphical small cancellation group contains a strongly contracting element.
• A geodesic $$\alpha$$ in a classical $$C'(1/6)$$ small cancellation group is Morse if and only if there is a sublinearly growing function $$f$$ such that for every relator $$\Pi$$ we have $$|\alpha\cap\Pi|\leq f(|\Pi|)$$. We construct uncountably many quasi-isometry classes of torsion-free, finitely generated groups such that every element is strongly contracting. The examples include Gromov monster groups. We show that when the defining graph has finite components then the translation lengths of group elements are rational and bounded away from 0. We also construct an example of a finitely generated group with a cyclic subgroup that is strongly contracting in one Cayley graph but not in another.

### Notes on the arXiv

Computing the Maximum Slope Invariant in Tubular Groups, preprint, 9pp, (2009).
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.
A Geometric Proof of the Structure Theorem for Cyclic Splittings of Free Groups, preprint, 11pp, (2012).
Application of the techniques of Papers 2 and 7 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.

All of my papers are also available on the arXiv.

### Other Stuff

Slides from Line Patterns talk, Joint Meetings 2011

### Teaching

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

### Thanks

I gratefully acknowledge support from:
(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 n°259527.
(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.