<p>Let <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(H\)</span> be groups and let <span class="SimpleMath">\(\varphi\)</span> and <span class="SimpleMath">\(\psi\)</span> be group homomorphisms from <span class="SimpleMath">\(H\)</span> to <span class="SimpleMath">\(G\)</span>. The pair <span class="SimpleMath">\((\varphi,\psi)\)</span> induces a (right) group action of <span class="SimpleMath">\(H\)</span> on <span class="SimpleMath">\(G\)</span> given by the map</p>
<p class="center">\[G \times H \to G \colon (g,h) \mapsto \varphi(h)^{-1} g\,\psi(h).\]</p>
<p>This group action is called <em><span class="SimpleMath">\((\varphi,\psi)\)</span>-twisted conjugation</em>. The orbits are called <em>Reidemeister classes</em> or <em>twisted conjugacy classes</em>, and the number of Reidemeister classes is called the <em>Reidemeister number</em> <span class="SimpleMath">\(R(\varphi,\psi)\)</span> of the pair <span class="SimpleMath">\((\varphi,\psi)\)</span>. The stabiliser of the identity <span class="SimpleMath">\(1_G\)</span> under the <span class="SimpleMath">\((\varphi,\psi)\)</span>-twisted conjugacy action of <span class="SimpleMath">\(H\)</span> is exactly the <em>coincidence group</em></p>
<p class="center">\[\operatorname{Coin}(\varphi,\psi) =
\left\{\, h \in H \mid \varphi(h) = \psi(h) \, \right\}.\]</p>
<p>Generalising this, the stabiliser of any <span class="SimpleMath">\(g \in G\)</span> is the coincidence group <span class="SimpleMath">\(\operatorname{Coin}(\iota_g\varphi,\psi)\)</span>, with <span class="SimpleMath">\(\iota_g\)</span> the inner automorphism of <span class="SimpleMath">\(G\)</span> that conjugates by <span class="SimpleMath">\(g\)</span>.</p>
<p>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 <span class="SimpleMath">\(X\)</span> and <span class="SimpleMath">\(Y\)</span> be topological spaces with universal covers <span class="SimpleMath">\(p \colon \tilde{X} \to X\)</span> and <span class="SimpleMath">\(q \colon \tilde{Y} \to Y\)</span> and let <span class="SimpleMath">\(\mathcal{D}(X), \mathcal{D}(Y)\)</span> be their covering transformations groups. Let <span class="SimpleMath">\(f,g \colon X \to Y\)</span> be continuous maps with lifts <span class="SimpleMath">\(\tilde{f}, \tilde{g} \colon \tilde{X} \to \tilde{Y}\)</span>. By <span class="SimpleMath">\(f_*\colon \mathcal{D}(X) \to \mathcal{D}(Y)\)</span>, denote the group homomorphism defined by <span class="SimpleMath">\(\tilde{f} \circ \gamma = f_*(\gamma) \circ \tilde{f}\)</span> for all <span class="SimpleMath">\(\gamma \in \mathcal{D}(X)\)</span>, and let <span class="SimpleMath">\(g_*\)</span> be defined similarly. The set of coincidence points <span class="SimpleMath">\(\operatorname{Coin}(f,g)\)</span> equals the union</p>
<p>For any two elements <span class="SimpleMath">\(\alpha, \beta \in \mathcal{D}(Y)\)</span>, the sets <span class="SimpleMath">\(p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g}))\)</span> and <span class="SimpleMath">\(p(\operatorname{Coin}(\tilde{f}, \beta \tilde{g}))\)</span> are either disjoint or equal. Moreover, they are equal if and only if there exists some <span class="SimpleMath">\(\gamma \in \mathcal{D}(X)\)</span> such that <span class="SimpleMath">\(\alpha = f_*(\gamma)^{-1} \circ \beta \circ g_*(\gamma)\)</span>, which is exactly the same as saying that <span class="SimpleMath">\(\alpha\)</span> and <span class="SimpleMath">\(\beta\)</span> are <span class="SimpleMath">\((f_*,g_*)\)</span>-twisted conjugate. Thus,</p>
<p>where <span class="SimpleMath">\([\alpha]\)</span> runs over the <span class="SimpleMath">\((f_*,g_*)\)</span>-twisted conjugacy classes. For sufficiently well-behaved spaces <span class="SimpleMath">\(X\)</span> and <span class="SimpleMath">\(Y\)</span> (e.g. nilmanifolds of equal dimension) we have that if <span class="SimpleMath">\(R(f_*,g_*) < \infty\)</span>, then</p>
<p>whereas if <span class="SimpleMath">\(R(f_*,g_*) = \infty\)</span> there exist continuous maps <span class="SimpleMath">\(f'\) and \(g'\)</span> homotopic to <span class="SimpleMath">\(f\)</span> and <span class="SimpleMath">\(g\)</span> respectively such that <span class="SimpleMath">\(\operatorname{Coin}(f',g') = \varnothing\)</span>.</p>
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