Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCounting Conjugacy Classes in $Out(F_N)$
138
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that if a f.g. group $G$ has a non-elementary WPD action on a hyperbolic metric space $X$, then the number of $G$-conjugacy classes of $X$-loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$. As an application we prove that for $Nge 3$ the number of distinct $Out(F_N)$-conjugacy classes of fully irreducibles $phi$ from an $R$-ball in the Cayley graph of $Out(F_N)$ with $loglambda(phi)$ on the order of $R$ grows exponentially in $R$.
rate research
Read More
$Out(F_n):=Aut(F_n)/Inn(F_n)$ denotes the outer automorphism group of the rank $n$ free group $F_n$. An element $phi$ of $Out(F_n)$ is polynomially growing if the word lengths of conjugacy classes in $F_n$ grow at most polynomially under iteration by $phi$. We restrict attention to the subset $UPG(F_n)$ of $Out(F_n)$ consisting of polynomially growing elements whose action on $H_1(F_n, Z)$ is unipotent. In particular, if $phi$ is polynomially growing and acts trivially on $H_1(F_n,Z_3)$ then $phi$ is in $UPG(F_n)$ and also every polynomially growing element of $Out(F_n)$ has a positive power that is in $UPG(F_n)$. In this paper we solve the conjugacy problem for $UPG(F_n)$. Specifically we construct an algorithm that takes as input $phi, psiin UPG(F_n)$ and outputs YES or NO depending on whether or not there is $thetain Out(F_n)$ such that $psi=thetaphitheta^{-1}$. Further, if YES then such a $theta$ is produced.
We give a short proof of Masbaum and Reids result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield-Thurston.
For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $delta$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.
We prove that $Out(F_N)$ is boundary amenable. This also holds more generally for $Out(G)$, where $G$ is either a toral relatively hyperbolic group or a finitely generated right-angled Artin group. As a consequence, all these groups satisfy the Novikov conjecture on higher signatures.
The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection from $T_n$ onto the symmetric group on $n$ symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in $T_n$, which quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of $z$-classes of involutions in $T_n$. We give a new proof of the structure of $Aut(T_n)$ for $n ge 3$, and show that $T_n$ is isomorphic to a subgroup of $Aut(PT_n)$ for $n geq 4$. Finally, we construct a representation of $T_n$ to $Aut(F_n)$ for $n ge 2$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
