Residual nilpotence and relations in free groups

Baumslag, Gilbert; Steinberg, Arthur

doi:10.1090/s0002-9904-1964-11123-x

Cited by 20 publications

(17 citation statements)

References 11 publications

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“…It follows that free groups are conjugacy separable, a result obtained by a number of authors, see [1], [5] and [6].…”

Section: Introductionmentioning

confidence: 72%

“…A group is conjugacy separable if each of its elements is conjugacy distinguished. Thus we obtain in particular yet another proof (see [1], [5] and [6] for others) of the conjugacy separability of free groups. Not that this leads to a better proof, but our result is stronger; that not every conjugacy separable group satisfies the conclusion of the above theorem is a triviality.…”

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confidence: 70%

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Conjugacy separating representations of free groups

Wehrfritz¹

1973

Proc. Amer. Math. Soc.

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Abstract.If G is a free group and g is an element of G we show that there exists a residually finite (commutative) integral domain R and a faithful matrix representation p of G over R of finite degree such that the conjugacy class of gp in Gp is closed in the topology induced on Gp by the Zariski topology on the full matrix algebra. If G is a free group and g is an element of G then our main result states that there exists a residually finite (commutative) integral domain R, an integer n and a faithful representation p of G into GL(n, R) such that the set gap of conjugates of gp in Gp is closed in Gp in the topology induced on Gp by the Zariski topology on the full matrix algebra Rn. (For a description of the Zariski topology see Chapter 5 of [8].)It is a simple lemma that any group with this property is conjugacy separable, where the latter concept is defined as follows. An element g of a group G is conjugacy distinguished in G if for each element x of G that is not conjugate to g there exists a homomorphism ofG into a finite group with X(p and g not conjugate in G. A group is conjugacy separable if each of its elements is conjugacy distinguished. Thus we obtain in particular yet another proof (see [1], [5] and [6] for others) of the conjugacy separability of free groups. Not that this leads to a better proof, but our result is stronger; that not every conjugacy separable group satisfies the conclusion of the above theorem is a triviality.Let G be any group. G can be made into a topological group by specifying the subgroups of G of finite index to be a base of the open neighbourhoods of the identity. We call this topology the profinite topology on G. It is well known, and very simple to show, that an element g of G is conjugacy distinguished in G if and only if ga is closed in the profinite topology. A Zariski topology on G induced by a faithful representation tends to be coarser than the profinite topology. Lemma 1. Let R be an integral domain and (a¿ :i e 1} a set of ideals ofRReceived by the editors October 10, 1972. AMS (MOS) subject classifications (1970). Primary 20E05, 20H20.

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“…It follows that free groups are conjugacy separable, a result obtained by a number of authors, see [1], [5] and [6].…”

Section: Introductionmentioning

confidence: 72%

mentioning

confidence: 70%

Conjugacy separating representations of free groups

Wehrfritz¹

1973

Proc. Amer. Math. Soc.

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“…Note that {x β 1 1 , x β 2 2 } is a basis of F . Therefore, in order to show that t is not a p-th power in F , it suffices to observe that t is not a p-th power in the free discrete group generated by {x β 1 1 , x β 2 2 } (see [3,Proposition 2]). Since σ x i (t) = α i ∈ pZ p for i = 1, 2, it follows from Proposition 5.6 that t is a test element of F .…”

Section: T -Test Arrangementsmentioning

confidence: 99%

Test elements in pro-p groups with applications in discrete groups

Snopce

Tanushevski

2017

Isr. J. Math.

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“…By 4. of Theorem 5.5, we can cut the CMQ vertex group π 1 ( M ) ≤ L along simple closed curves to get a new splitting with Bass-Serre tree T 1 such that T 1 T JSJ is obtained by perhaps collapsing edges dual to the simple closed curves and there is an L-equivariant continuous (but perhaps not simplicial) map T 1 T . The subgroup π 1 ( ) is a vertex group of T 1 , and in particular the element h ∈ π 1 ( ) acts elliptically on T 1 .…”

Section: Lemma 56 the Splitting Lmentioning

confidence: 99%