Normal Forms for Automorphisms of Universal Coxeter Groups and Palindromic Automorphisms of Free Groups

Piggott, Adam; Ruane, Kim

doi:10.1142/s0218196710006035

Cited by 2 publications

(3 citation statements)

References 13 publications

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“…Using geometric techniques, Glover and Jensen [5] proved this conjecture and also calculated the virtual cohomological dimension of ΠA n . Using methods from logic theory, Piggott and Ruane [12] constructed Markov languages of normal forms for ΠA n . The pure palindromic automorphism group is defined as P ΠA n = µ ij , t k | 1 ≤ i = j ≤ n and 1 ≤ k ≤ n .…”

Section: The Groupmentioning

confidence: 99%

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Palindromic automorphisms of free groups

2015

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Abstract. Let Fn be the free group of rank n with free basis X = {x1, . . . , xn}. A palindrome is a word in X ±1 that reads the same backwards as forwards. The palindromic automorphism group ΠAn of Fn consists of those automorphisms that map each xi to a palindrome. In this paper, we investigate linear representations of ΠAn, and prove that ΠA2 is linear. We obtain conjugacy classes of involutions in ΠA2, and investigate residual nilpotency of ΠAn and some of its subgroups. Let IAn be the group of those automorphisms of Fn that act trivially on the abelianisation, P In be the palindromic Torelli group of Fn, and EΠAn be the elementary palindromic automorphism group of Fn. We prove that P In = IAn ∩ EΠA ′ n . This result strengthens a recent result of Fullarton [3].

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Section: The Groupmentioning

confidence: 99%

“…Clearly, IA n ∩ EΠA ′ n ⊂ IA n ∩ ΠA n = P I n . In [3], Fullarton showed that P I n is normally generated in ΠA n by the automorphisms [µ 12 3 and hence they also lie in IA n ∩EΠA ′ n . Note that, IA n is a normal subgroup of Aut(F n ).…”

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confidence: 99%

Palindromic automorphisms of free groups

2015

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“…Extending this work in [13], Jensen, McCommand and Meier computed the Euler characteristic of ΠA(F n ) and EΠA(F n ). In [16], Piggott and Ruane constructed Markov languages of normal forms for ΠA(F n ) using methods from logic theory. In [18,19], Nekritsukhin investigated some basic group theoretic questions about ΠA(F n ).…”

Section: Introductionmentioning

confidence: 99%

Palindromic automorphisms of free nilpotent groups

Bardakov

Gongopadhyay

Нещадим

et al. 2017

Journal of Pure and Applied Algebra

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Abstract. In this paper, we initiate the study of palindromic automorphisms of groups that are free in some variety. More specifically, we define palindromic automorphisms of free nilpotent groups and show that the set of such automorphisms is a group. We find a generating set for the group of palindromic automorphisms of free nilpotent groups of step 2 and 3. In particular, we obtain a generating set for the group of central palindromic automorphisms of these groups. In the end, we determine central palindromic automorphisms of free nilpotent groups of step 3 which satisfy the necessary condition of Bryant-Gupta-Levin-Mochizuki for a central automorphism to be tame.

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Normal Forms for Automorphisms of Universal Coxeter Groups and Palindromic Automorphisms of Free Groups

Cited by 2 publications

References 13 publications

Palindromic automorphisms of free groups

Palindromic automorphisms of free groups

Palindromic automorphisms of free nilpotent groups

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