Searcharxiv⌕ Search

arXiv subjects

Alexandra Pettet

Publications and source records attributed to Alexandra Pettet.

15 recordsLinked to original sources
arXiv2014

Ergodic decompositions for folding and unfolding paths in Outer space

We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the folding/unfolding path. It follows that non-unique ergodicity leads to distortion when projecting to the complex of free factors, and we give two applications of this fact. First, we show that if a subgroup $H$ of $Out(\FN)$ quasi-isometrically embeds into the complex of free factors via the orbit map, then the limit set of $H$ in the boundary of Outer space consists of trees that are uniquely ergodic and have uniquely ergodic dual lamination. Second, we describe the Poisson boundary for random walks coming from distributions with finite first moment with respect to the word metric on $Out(\FN)$: almost every sample path converges to a tree that is uniquely ergodic and that has a uniquely ergodic dual lamination, and the corresponding hitting measure on the boundary of Outer space is the Poisson boundary. This improves a recent result of Horbez. We also obtain sublinear tracking of sample paths with Lipschitz geodesic rays.

math.GT↗
arXiv2013

Relative shapes of thick subsets of moduli space

A closed hyperbolic surface of genus $g\ge 2$ can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly 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 {\em trivalent}. In this paper, in a first attempt to understand the "shape" of the subset $\ts$ of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in g, with the set $\tri$ of trivalent surfaces. As our main result, we find that the set $\ts \cap \tri$ is metrically "sparse" in $\ts$ (where we equip $\moduli$ with either the Thurston or the Teichmüller metric).

math.GT↗
arXiv2012

Relative twisting in Outer space

Subsurface projection has become indispensable in 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.

math.GR↗
arXiv2010

Small filling sets of curves on a surface

We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus $g$ which fill and pairwise intersect at most $K\ge 1$ times is $2\sqrt{g}/\sqrt{K}$ as $g \to \infty$ . We then bound from below the cardinality of a filling set of systoles by $g/\log(g)$. This illustrates that the topological condition that a set of curves pairwise intersect at most once is quite far from the geometric condition that such a set of curves can arise as systoles.

math.GT↗
arXiv2010

On the Andreadakis-Johnson filtration of the automorphism group of a free group

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.

math.GR↗
arXiv2010

On the fundamental group of Hom(Z^k,G)

Let G be a compact Lie group, and consider the variety Hom(Z^k,G) of representations of Z^k into G. We view this as a based space by designating the trivial representation to be its base point. We prove that the fundamental group of this space is naturally isomorphic to π_1(G)^k.

math.AT↗
arXiv2010

Current twisting and nonsingular matrices

We show that for k at least 3, 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 Z^k is the given matrix.

math.GR↗
arXiv2010

Twisting out fully irreducible automorphisms

By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves that fill, every element not conjugate to a power of one of the twist is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.

math.GR↗
arXiv2009

Finiteness properties for a subgroup of the pure symmetric automorphism group

Let F_n be the free group on n generators, and PΣ_n be the group of automorphisms of F_n which send each generator to a conjugate of itself. Let K_n be the kernel of the homomorphism from PΣ_n to PΣ_{n-1} induced by mapping one of the free group generators to the identity. We show that K_n has cohomological dimension n-1, and that the ith cohomology groups are infinitely generated for all i between 2 and n-1. It follows that K_n is not finitely presentable for n>2.

math.GR↗
arXiv2009

Periodic maximal flats are not peripheral

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 can be homotoped out of K.

math.GT↗
arXiv2007

The spine which was no spine

Let T_n be the Teichmueller space of flat metrics on the n-dimensional torus and identify SL(n,Z) with the corresponding mapping class group. We prove that the subset Y consisting of those points at which the systoles generate the fundamental group of the torus is, for n > 4, not contractible. In particular, Y is not an SL(n,Z)-equivariant deformation retract of T_n.

math.GT↗
arXiv2005

The Johnson homomorphism and the second cohomology of IA_n

Let F_n be the free group on n generators. Define IA_n to be group of automorphisms of F_n that act trivially on first homology. The Johnson homomorphism in this setting is a map from IA_n to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of IA_n. A descending central series of IA_n is given by the subgroups K_n^(i) which act trivially on F_n/F_n^(i+1), the free rank n, degree i nilpotent group. It is a conjecture of Andreadakis that K_n^(i) is equal to the lower central series of IA_n; indeed K_n^(2) is known to be the commutator subgroup of IA_n. We prove that the quotient group K_n^(3)/IA_n^(3) is finite for all n and trivial for n=3. We also compute the rank of K_n^(2)/K_n^(3).

math.GR↗