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Let $G$ be a group (if it helps, assume that $G$ is a Lie group or finite). Is a pair of elements $(g, h) \in G \times G$ determined up to simultaneous conjugacy by the conjugacy class of every element $w(g, h) \in G$, where $w$ runs over all words in the free group on two generators?

If $G$ is finite, can we bound the length of the words $w$ needed in terms of $|G|$?

If the answer to the above question is positive, let $\pi$ be a second group (if it helps, assume that $\pi$ is finitely presented). $G$ acts on the set $\text{Hom}(\pi, G)$ by pointwise conjugation. Is an element $\phi \in \text{Hom}(\pi, G)$ determined up to conjugacy by the conjugacy class of every element $\phi(w)$ where $w \in \pi$? (The above is the special case $\pi = F_2$.)

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2 Answers 2

up vote 7 down vote accepted

For a concrete example consider the symmetric group on 6 symbols. The pairs ((1,2)(3,4),(1,3)(2,4)) and ((1,2)(3,4),(3,4)(5,6)) both generate Klein 4-groups. Words in them are either trivial of conjugate to (1,2)(3,4). The generated subgroups (and thus the pairs) are not conjugate as one fixes two points.

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Suppose a group $G$ had the property that n-tuples $\lbrace x_1,\dots, x_n\rbrace$ and $\lbrace y_1,\dots,y_n\rbrace$ satisfy: if $w(x_1,\dots,x_n)$ is conjugate to $w(y_1,\dots,y_n)$ for all $w\in F_n$ then there is a uniform conjugator $g$ so that $y_i=gx_ig^{-1}$. Then as a corollary you get that if an endomorphism of $G$, satisfies $\varphi(x)$ is conjugate to $x$ for all $x$ then $\varphi$ is an inner automorphism.

However there are non-examples to this property in several classes of groups, including finite groups. The original property does however hold for torsion-free $\delta$-hyperbolic groups. This is proved in the paper "On endomorphisms of torsion-free hyperbolic groups", which also has references to the previous work.

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Interesting! So I guess $\pi$ can't have $F_2$ as a quotient group. Do you know what happens if $\pi$ is, say, the fundamental group of a closed surface? –  Qiaochu Yuan Oct 4 '12 at 7:41
@Qiaochu: Most of those are hyperbolic! –  Steve D Oct 4 '12 at 22:37
@Steve: my understanding is that the result Gjergji describes is about examples of $G$ such that the result above fails for $\pi = F_2$. I want to change $\pi$ to a different group and ask if the result is still true for all $G$ in that case. –  Qiaochu Yuan Oct 5 '12 at 1:21
@Qiaochu: ah ok, I thought you were asking if the original question was true for surface groups. –  Steve D Oct 5 '12 at 4:43

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