Translator Disclaimer
March 2018 Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties
Matthew C. B. Zaremsky
Michigan Math. J. 67(1): 133-158 (March 2018). DOI: 10.1307/mmj/1516330971

Abstract

The BNSR-invariants of a group G are a sequence Σ1(G)Σ2(G) of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAutn and pure symmetric automorphism group PΣAutn of the free group Fn and inspect their BNSR-invariants. We prove that for n2, all the “positive” and “negative” character classes of PΣAutn lie in Σn2(PΣAutn)Σn1(PΣAutn). We use this to prove that for n2, Σn2(ΣAutn) equals the full character sphere S0 of ΣAutn but Σn1(ΣAutn) is empty, so in particular the commutator subgroup ΣAutn' is of type Fn2 but not Fn1. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

Citation

Download Citation

Matthew C. B. Zaremsky. "Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties." Michigan Math. J. 67 (1) 133 - 158, March 2018. https://doi.org/10.1307/mmj/1516330971

Information

Received: 22 August 2016; Revised: 16 May 2017; Published: March 2018
First available in Project Euclid: 19 January 2018

zbMATH: 06965593
MathSciNet: MR3770857
Digital Object Identifier: 10.1307/mmj/1516330971

Subjects:
Primary: 20F65
Secondary: 20F28, 57M07

Rights: Copyright © 2018 The University of Michigan

JOURNAL ARTICLE
26 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.67 • No. 1 • March 2018
Back to Top