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Suppose $G$ is a finitely presented group with generators $a_1, \ldots, a_n$. Suppose $f \colon G \to G$ is a group endomorphism specified by defining $f(a_1), \ldots, f(a_n)$. As expected, we define a fixed point of $f$ to be any element $g \in G$ such that $f(g) = g$ and, as $f(\mathop{id}) = \mathop{id}$, we say that $\mathop{id}$ is the trivial fixed point.

For example, let $G = \langle a | \rangle$ and $f$ and $g$ be defined by $f(a) = \mathop{id}$ and $g(a) = a^2$. Note in both cases $f$ and $g$ have no non-trivial fixed points and for this particular group we can determine that an endomorphism $f$ has a non-trivial fixed point if and only if $f(a) = a$.

For what groups is it possible to determine whether or not any given endomorphism has a non-trivial fixed point?

I am particularly interested in the question of:

Is $\langle a, b, c | \rangle$ such a group?

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For the free group an algorithm is here: Sykiotis, Mihalis Fixed points of symmetric endomorphisms of groups. Internat. J. Algebra Comput. 12 (2002), no. 5, 737–745.

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This provides an answer to a related question: can one determine whether an endomorphism of a free group maps a cyclic subgroup into a conjugate? Perhaps it can be improved to answer your second question exactly.

By the Combination Theorem for hyperbolic groups (Bestvina and Feighn), we have the following.

Theorem: Let $F$ be a finitely generated free group and let $\phi:F\to F$ be an endomorphism. The following are equivalent:

  1. $\phi$ maps a non-trivial cyclic subgroup into a conjugate;
  2. the ascending HNN extension $\Gamma_\phi=F*_\phi$ is not word-hyperbolic.

Now, Panos Papasoglu described an algorithm that confirms if a given presentation defines a word-hyperbolic group. His algorithm doesn't terminate if it doesn't.

On the other hand, given $g\in F$ and an integer $k$, the solution to the conjugacy problem in $F$ determines whether or not $\phi(g)$ is conjugate to $g^k$. Therefore, a naive enumeration of elements of $F$ and integers will eventually determine if $\phi$ maps a non-trivial cyclic subgroup into a conjugate.

Running these two procedures in parallel, one eventually determines if $\phi$ maps a cyclic subgroup into a conjugate. Of course, this algorithm is completely impractical. It would be interesting to know if a more efficient algorithm exists.

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