arXiv subjects
Publications and source records attributed to Nic Koban.
We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.
In 1987 Bieri, Neumann and Strebel introduced a geometric invariant for discrete groups. In this article we compute and explicitly describe the BNS-invariant for the pure braid groups.
In this paper, we compute the Σ^n(G) and Ω^n(G) invariants when 1 \rightarrow H \rightarrow G \rightarrow K \rightarrow 1 is a short exact sequence of finitely generated groups with K finite. We also give sufficient conditions for G to have the R_{\infty} property in terms of Ω^n(H) and Ω^n(K) when either K is finite or the sequence splits. As an application, we construct a group F \rtimes? Z_2 where F is the R. Thompson's group F and show that F \rtimes Z_2 has the R_{\infty} property while F is not characteristic.
In this note, we compute the Σ^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a short exact sequence of finitely generated groups with K finite. As an application, we construct a group F semidirect Z_2 where F is the R. Thompson's group F and show that F semidirect Z_2 has the R-infinity property while F is not characteristic. Furthermore, we construct a finite extension G with finitely generated commutator subgroup G' but has a finite index normal subgroup H with infinitely generated H'.
We compute the Ω^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a split short exact sequence. We use this result to compute the invariant for pure and full braid groups on compact surfaces. Applications to twisted conjugacy classes and to finite generation of commutator subgroups are also discussed.
A group $G$ is said to have the property $R_\infty$ if every automorphism $ϕ\in {\rm Aut}(G)$ has an infinite number of $ϕ$-twisted conjugacy classes. Recent work of Gonçalves and Kochloukova uses the $Σ^n$ (Bieri-Neumann-Strebel-Renz) invariants to show the $R_{\infty}$ property for a certain class of groups, including the generalized Thompson's groups $F_{n,0}$. In this paper, we make use of the $Ω^n$ invariants, analogous to $Σ^n$, to show $R_{\infty}$ for certain finitely generated groups. In particular, we give an alternate and simpler proof of the $R_{\infty}$ property for BS(1,n). Moreover, we give examples for which the $Ω^n$ invariants can be used to determine the $R_{\infty}$ property while the $Σ^n$ invariants techniques cannot.