Research interests:
geometric group theory and topology; more specifically, the mapping class group and the outer automorphism group of a free group, and the spaces they act on.

A more detailed research statement will be posted soon.


In progress:
  • Abstract commensurators of the Johnson filtration of the mapping class group, (with Martin Bridson and Juan Souto).

    Abstract: The Johnson filtration consists of the subgroups of the mapping class group of a surface acting trivially on nilpotent quotients of the surface group. The first of these subgroups is the well-known Torelli group, which acts trivially on the first homology group of the surface. We prove that the abstract commensurator of every subgroup of the Johnson filtration is the extended mapping class group, for surfaces of genus at least 7.

  • Ergodic decompositions for folding and unfolding paths in outer space, (with Hossein Namazi and Patrick Reynolds).

    Abstract: We provide an analogue of Masur's criterion for a folding or unfolding path in Culler-Vogtmann's outer space. Our result is similar to a result of McMullen in which the number of homothetically distinct ergodic measures on the vertical foliation of a Teichmueller ray converging to a noded surface in Mumford's compactification is equal to the number of components in the complement of the nodes.


Papers:
  • An algorithm to detect full irreducibility by bounding the volume of periodic free factors, (with Matt Clay and Johanna Mangahas), submitted. [pdf]

    Abstract: We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.

  • Relative shapes of thick subsets of moduli space, (with Jim Anderson and Hugo Parlier), submitted. [pdf]

    A hyperbolic surface of genus g can be decomposed into pants along shortest possible curves, and if these curves are sufficiently short (but bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. These combinatorics are determined by a trivalent graph, so we call such surfaces trivalent. In this paper, in a first attempt to understand the "shape" of the subset X_g of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in g, with the set Y_g of trivalent surfaces. As our main result, we find that the set X_g in Y_g is generically "sparse" in X_g. We consider the moduli space M_g equipped with either the Teichmueller or the Thurston metric.

  • Relative twisting in Outer space, (with Matt Clay),
    J. Topol. Anal. 4 (2012), 173-201 [ J. Topol. Anal. ]

    Abstract: Subsurface projection has become indispensable to studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the thin part of Culler-Vogtmann's Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.

  • Commuting tuples in reductive groups and their maximal compact subgroups, (with Juan Souto), to appear in Geom. Topol. [ pdf ]

    Abstract: Let G be a connected reductive algebraic group and K in G a maximal compact subgroup. We consider the representation spaces Hom(Z^k,K) and Hom(Z^k,G) with the topology induced from an embedding into K^k and G^k, respectively. The goal of this paper is to prove that Hom(Z^k,K) is a strong deformation retract of Hom(Z^k,G).

  • On the fundamental group of Hom(Z^k,G), (with José Manuel Gómez and Juan Souto),
    Math. Z. 271 (2012), no. 1-2, 33-44. [
    Math. Z. ].

    Abstract: Let G be a compact Lie group, and consider the variety Hom(Z^k,G) of representations of Z^k into G. The goal of this paper is to prove that the fundamental group of Hom(Z^k,G) is naturally isomorphic to the direct product of k copies of the fundamental group of G.

  • On the Johnson filtration of the automorphism group of a free group, (with Fred Cohen and Aaron Heap ),
    J. Algebra, Vol. 329, Issue 1 (2011), 72-91. [ J. Algebra ]

    Abstract: The Johnson filtration of the automorphism group of a free group is composed of those automorphisms which act trivially on nilpotent quotients of the free group. We compute cohomology classes as follows: (i) we analyze analogous classes for a subgroup of the pure symmetric automorphism group of a free group, and (ii) we analyze features of these classes which are preserved by the Johnson homomorphism. One consequence is that the ranks of the cohomology groups in any fixed dimension between 1 and n-1 increase without bound for terms deep in the Johnson filtraton.

  • Periodic maximal flats are not peripheral, (with Juan Souto), to appear in J. Topol. [ pdf ]

    Abstract: We prove that every non-positively curved locally symmetric manifold M of finite volume contains a compact set K such that no periodic maximal flat in M can be homotoped out of K.

  • Small filling sets of curves on a surface, (with Jim Anderson and Hugo Parlier),
    Topology Appl. 158 (2011) 84-92. [ pdf ]

    Abstract: We consider a set of simple closed curves on a surface of genus g which fill the surface and which pairwise intersect at most once. We show the asymptotic growth rate of the smallest number in such a set is proportional to the square root of g as g goes to infinity. We then bound from below the cardinality of a filling set of systoles (shortest simple closed curves) on a hyperbolic surface by g/log(g). The topological condition that set of curves pairwise intersect at most once is thus quite far from the geometric condition that set of curves can arise as systoles.

  • Current twisting and nonsingular matrices, (with Matt Clay),
    Comm. Math. Helv. 87 (2012), 385-407. [ Comm. Math. Helv. ]

    Abstract: We show that for k at least 3, that given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on the rank k abelian group is the given matrix.

  • Twisting out fully irreducible automorphisms, (with Matt Clay),
    Geom. Funct. Anal., Vol. 20, Issue 3 (2010), 657-689. [GAFA ]

    Abstract: By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two simple closed curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a rank k non-abelian free group.

  • Finiteness properties for a subgroup of the pure symmetric automorphism group,
    C. R. Acad. Sci. Paris, Ser. I 348 (2010) 127-130. [ C.R. Acad. Sci. Paris ]

    Abstract: We consider the pure symmetric automorphism group of the free group which sends every generator to a conjugate of itself. This group corresponds to the mapping class group of the complement in 3-space of the trivial link of circles, and is therefore analogous to the pure braid group. By a theorem of Collins-Gilbert, the kernel K of the map induced by "forgetting" a generator is a finitely generated group which is not finitely presentable (in contrast to the "forgetting" map for the pure braid group where the kernel is a finite rank nonabelian free group). We give a geometric proof of this result which furthermore shows that K has cohomological dimension n-1, and that the degree d homology group is not finitely generated for d between 2 and n-1.

  • Minimality of the well-rounded retract, (with Juan Souto),
    Geom. Topol., Volume 12 (2008), 1543-1556. [
    G&T ]

    Abstract: We prove that the well-rounded retract of the symmetric space for SL(n,Z), also known as the Teichmueller Space of flat metrics on the n-dimensional torus, is a minimal invariant spine.

  • The spine that was no spine, (with Juan Souto),
    Enseign. Math. 54 (2008), 273-285. [
    Enseign. Math. ]

    Abstract: We consider the Teichmueller Space of flat metrics on the n-dimensional torus and prove that the subset consisting of those points whose systoles generate the fundamental group of the torus is, for n at least 5, not contractible. In particular, this subset is not a deformation retract of the Teichmueller Space.

  • The Johnson homomorphism and the second cohomology of IA_n,
    Algebr. Geom. Topol., Volume 5 (2005), 725-740. [ AGT]

    Abstract: Implementing methods from the classical representation theory of GL(n,C), we compute cohomlogy classes for the subgroup IA_n of the automorphism group Aut(F) of a free group which acts trivially on homology; in particular, we compute the kernel of cup product on its first cohomology. We also achieve some results relating the Johnson filtration of Aut(F) to the lower central series of IA_n for low degree terms.

  • Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra, (with Thomas J. Haines ),
    J. Algebra, vol. 252, (2002), 127-149. [ pdf ]



My research is partially supported by an NSERC Discovery Grant.