Fixed Subgroups are Compressed in Free Groups

Abstract

In this paper, we prove that the fixed subgroup of an arbitrary family of endomorphisms ψ _i , i ∈ I, of a finitely generated free group F, is F-super-compressed. In particular, r(∩ _i∈I Fix ψ _i ) ≤ r(M) for every subgroup M ≤ F containing ∩ _i∈I Fix ψ _i . This provides new evidence towards the inertia conjecture for fixed subgroups of free groups. As a corollary, we show that, in the free group of rank n, every strictly ascending chain of fixed subgroups has length at most 2n. This answers a question of Levitt.

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Acknowledgments

Both authors thank Warren Dicks for interesting comments and suggestions for improving some aspects of the paper. The first named author gratefully acknowledges the postdoctoral grant SB2001-0128 funded by the Spanish government, and thanks the CRM for its hospitality during the last period of this research. The second named author gratefully acknowledges partial support by DGI (Spain) through grant BFM2000-0354.

Notes

^#Communicated by A. Facchini.