Research interests:
geometric group theory and topology; more specifically, the
mapping class group and the outer automorphism group of a free group, together
with
the
spaces they act on.
More details are in my
research statement .
In preparation:
- On the Johnson filtration of the automorphism group of a
free group, (with Fred Cohen
and Aaron Heap ), in
preparation.
Abstract: The Johnson filtration of the automoprhism 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 d
between 1 and n-1 for the increase without bound for terms deep in the
Johnson filtraton.
Papers:
- Periodic maximal flats are not peripheral, (with
Juan Souto ),
submitted.
[ 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), submitted.
[ 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 ), submitted.
[ pdf
]
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
), submitted. [ pdf
]
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, submitted. [ pdf
]
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
),
Geometry and Topology, 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
),
L'Enseignement Mathématique 54 (2008), 273-285.
[ pdf ]
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,
Algebraic and Geometric Topology, 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 ]
[ ps ]
[ dvi ]
My research is partially supported by NSF grant DMS-0856143 and NSF
RTG grant DMS-0602191.