Conjugacy separating representations of free groups

Abstract

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 <f>ofG into a finite group with X(p and g<f> 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.

The linearity of wreath products

“…Consequently (A"')' centralizes B/[B, N] and hence also B/f] N[B, N]. Now A' is a finitely generated linear group over F. In such a group Zariskiclosed subgroups are profinitely closed, see[7] Lemma 2. If coeQ, then C K (a>) = C K (A W ).…”

Decision problems, complexity, traces, and representations

“…In this subsection, we prove Theorem 1.6. The construction of the representation needed to verify Theorem 1.6 in the case of free groups follows Wehrfritz [68]. The surface group case is similar.…”

“…Property (B). Following a construction of Wehrfritz [68] for free groups, we can prove that finitely generated free groups and surface groups have (B). Theorem 1.6.…”

Separating Conjugates in Free-by-Finite Groups

“…It has been shown by Stebe [11], as well as by Remeslennikov [8] and Wehrfritz [12] using different techniques, that the elements of infinite order in a free-by-finite group are conjugacy distinguished. Moreover, free-by-finite groups appear in a nice way as "fundamental groups" of graphs of groups (cf.…”