M 1717-N25 Geometric and Analytic Aspects of Free Group Automorphisms
Project Publications
Further Activities
Final Report

a) Peer-reviewed

Growth Tight Actions w/ Goulnara N. Arzhantseva and Jing Tao, Pacific Journal of Mathematics, 278 (2015), no. 1, 1–49. Open Access logo
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
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
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
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
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, in press, 43pp. Open Access logo
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.
Characterizations of Morse Quasi-geodesics via Superlinear Divergence and Sublinear Contraction w/ Goulnara N. Arzhantseva, Dominik Gruber, and David Hume, submitted, 29pp. preprint
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.
Contracting Geodesics in Graphical Small Cancellation Groups w/ Goulnara N. Arzhantseva, Dominik Gruber, and David Hume, submitted, 41pp. preprint
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.


b) Non peer-reviewed


c) Stand-alone publications


d) Publications for the general public and other publications



With support from
FWFDer Wissenschaftsfonds