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.
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 ]
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).
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.
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.
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.
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
[ 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.
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.
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.
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.
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My research is partially supported by an NSERC Discovery Grant.