Farrell-Jones Conjecture for free-by-cyclic groups
We prove the Farrell-Jones conjecture for free-by-cyclic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture.
math.GT↗
arXiv subjects
Derrick Wigglesworth
Publications and source records attributed to Derrick Wigglesworth.
The Farrell-Jones Conjecture for hyperbolic-by-cyclic groups
We prove the Farrell-Jones Conjecture for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture by Bartels-Lück-Reich, as well as the structure theory of mapping tori by Dahmani-Krishna.
math.GT↗
Large lower bounds for the betti numbers of graded modules with low regularity
Suppose that $M$ is a finitely-generated graded module of codimension $c\geq 3$ over a polynomial ring and that the regularity of $M$ is at most $2a-2$ where $a\geq 2$ is the minimal degree of a first syzygy of $M$. Then we show that the sum of the betti numbers of $M$ is at least $β_0(M)(2^c + 2^{c-1})$. In addition, if $c \geq 9$ then for each $1\leq i\leq \lceil c/2\rceil$, we show $β_i(M)\geq 2β_0(M){c \choose i}$.
math.AC↗
Loxodromic elements in the cyclic splitting complex and their centralizers
We show that an outer automorphism acts loxodromically on the cyclic splitting complex if and only if it has a filling lamination and no generic leaf of the lamination is carried by a vertex group of a cyclic splitting. This is the analog for the cyclic splitting complex of Handel-Mosher's theorem on loxodromics for the free splitting complex. We also show that such outer automorphisms have virtually cyclic centralizers.
math.GR↗
Distortion for Abelian Subgroups of $\mathrm{Out}(F_n)$
We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for $\mathrm{Out}(F_n)$.
math.GR↗