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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