Fixed Subgroups are Compressed in Free Groups
Fixed Subgroups are Compressed in Free Groups
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DOI：
10.1081/agb-200027786
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发表时间：
2004-12
影响因子：
0.7
通讯作者：
A. Martino;E. Ventura
中科院分区：
文献类型：
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作者：
A. Martino;E. Ventura
本文证明了一个自由群F的任意自同态族的不动点子群是F-超压缩的。特别地，对于每个包含IFix i的子群M ≤ F，r（i ∈ IFix i）≤ r（M）。这为自由群的不动子群的惯性猜想提供了新的证据。作为推论，我们证明了在秩为n的自由群中，每个严格升的固定子群链的长度不超过2n。这回答了莱维特的一个问题。
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.