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Symmetric automorphisms of free groups, BNSR-invariants, and finiteness properties. (English) Zbl 06965593

Summary: The BNSR-invariants of a group \(G\) are a sequence \(\Sigma^{1}(G)\supseteq \Sigma^{2}(G)\supseteq \cdots \) of geometric invariants that reveal important information about finiteness properties of certain subgroups of \(G\). We consider the symmetric automorphism group \(\operatorname{\Sigma Aut}_{n}\) and pure symmetric automorphism group \(\operatorname{P\!\operatorname{\Sigma Aut}}_{n}\) of the free group \(F_{n}\) and inspect their BNSR-invariants. We prove that for \(n\geq 2\), all the “positive” and “negative” character classes of \(\operatorname{P\!\operatorname{\Sigma Aut}}_{n}\) lie in \(\Sigma^{n-2}(\operatorname{P\!\operatorname{\Sigma Aut}}_{n})\setminus \Sigma^{n-1}(\operatorname{P\!\operatorname{\Sigma Aut}}_{n})\). We use this to prove that for \(n\geq 2\), \(\Sigma^{n-2}(\operatorname{\Sigma Aut}_{n})\) equals the full character sphere \(S^{0}\) of \(\operatorname{\Sigma Aut}_{n}\) but \(\Sigma^{n-1}(\operatorname{\Sigma Aut}_{n})\) is empty, so in particular the commutator subgroup \(\operatorname{\Sigma Aut}_{n}'\) is of type \(\operatorname{F}_{n-2}\) but not \(\operatorname{F}_{n-1}\). Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

MSC:

20-XX Group theory and generalizations
20F65 Geometric group theory
20F28 Automorphism groups of groups
57M07 Topological methods in group theory
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