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<h3>2 Mathematical background</h3>

Let \(G\) and \(H\) be groups and let \(\varphi\) and \(\psi\) be group homomorphisms from \(H\) to \(G\). The pair \((\varphi,\psi)\) induces a (right) group action of \(H\) on \(G\) given by the map

\[G \times H \to G \colon (g,h) \mapsto \varphi(h)^{-1} g\,\psi(h).\]

This group action is called \((\varphi,\psi)\)-twisted conjugation. The orbits are called Reidemeister classes or twisted conjugacy classes, and the number of Reidemeister classes is called the Reidemeister number \(R(\varphi,\psi)\) of the pair \((\varphi,\psi)\). The stabiliser of the identity \(1_G\) under the \((\varphi,\psi)\)-twisted conjugacy action of \(H\) is exactly the coincidence group

\[\operatorname{Coin}(\varphi,\psi) =
\left\{\, h \in H \mid \varphi(h) = \psi(h) \, \right\}.\]

Generalising this, the stabiliser of any \(g \in G\) is the coincidence group \(\operatorname{Coin}(\iota_g\varphi,\psi)\), with \(\iota_g\) the inner automorphism of \(G\) that conjugates by \(g\).

Twisted conjugacy originates in Reidemeister-Nielsen fixed point and coincidence theory, where it serves as a tool for studying fixed and coincidence points of continuous maps between topological spaces. Below, we briefly illustrate how and where this algebraic notion arises when studying coincidence points. Let \(X\) and \(Y\) be topological spaces with universal covers \(p \colon \tilde{X} \to X\) and \(q \colon \tilde{Y} \to Y\) and let \(\mathcal{D}(X), \mathcal{D}(Y)\) be their covering transformations groups. Let \(f,g \colon X \to Y\) be continuous maps with lifts \(\tilde{f}, \tilde{g} \colon \tilde{X} \to \tilde{Y}\). By \(f_*\colon \mathcal{D}(X) \to \mathcal{D}(Y)\), denote the group homomorphism defined by \(\tilde{f} \circ \gamma = f_*(\gamma) \circ \tilde{f}\) for all \(\gamma \in \mathcal{D}(X)\), and let \(g_*\) be defined similarly. The set of coincidence points \(\operatorname{Coin}(f,g)\) equals the union

\[\operatorname{Coin}(f,g) = \bigcup_{\alpha \in \mathcal{D}(Y)}
p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g})).\]

For any two elements \(\alpha, \beta \in \mathcal{D}(Y)\), the sets \(p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g}))\) and \(p(\operatorname{Coin}(\tilde{f}, \beta \tilde{g}))\) are either disjoint or equal. Moreover, they are equal if and only if there exists some \(\gamma \in \mathcal{D}(X)\) such that \(\alpha = f_*(\gamma)^{-1} \circ \beta \circ g_*(\gamma)\), which is exactly the same as saying that \(\alpha\) and \(\beta\) are \((f_*,g_*)\)-twisted conjugate. Thus,

\[\operatorname{Coin}(f,g) = \bigsqcup_{[\alpha]}
p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g})),\]

where \([\alpha]\) runs over the \((f_*,g_*)\)-twisted conjugacy classes. For sufficiently well-behaved spaces \(X\) and \(Y\) (e.g. nilmanifolds of equal dimension) we have that if \(R(f_*,g_*) < \infty\), then

\[R(f_*,g_*) \leq \left|\operatorname{Coin}(f,g)\right|,\]

whereas if \(R(f_*,g_*) = \infty\) there exist continuous maps \(f'\) and \(g'\) homotopic to \(f\) and \(g\) respectively such that \(\operatorname{Coin}(f',g') = \varnothing\).

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