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<div class="ChapSects"><a href="chap2.html#X7EF1B6708069B0C7">2 <span class="Heading">Mathematical background</span></a>
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<h3>2 <span class="Heading">Mathematical background</span></h3>

<p>Let <span class="Math">G</span> and <span class="Math">H</span> be groups and let <span class="Math">\varphi</span> and <span class="Math">\psi</span> be group homomorphisms from <span class="Math">H</span> to <span class="Math">G</span>. The pair <span class="Math">(\varphi,\psi)</span> induces a (right) group action of <span class="Math">H</span> on <span class="Math">G</span> given by the map</p>

<p class="pcenter">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="Math">(\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="Math">R(\varphi,\psi)</span> of the pair <span class="Math">(\varphi,\psi)</span>. The stabiliser of the identity <span class="Math">1_G</span> under the <span class="Math">(\varphi,\psi)</span>-twisted conjugacy action of <span class="Math">H</span> is exactly the <em>coincidence group</em></p>

<p class="pcenter">\operatorname{Coin}(\varphi,\psi) =
 \left\{\, h \in H \mid \varphi(h) = \psi(h) \, \right\}.</p>

<p>Generalising this, the stabiliser of any <span class="Math">g \in G</span> is the coincidence group <span class="Math">\operatorname{Coin}(\iota_g\varphi,\psi)</span>, with <span class="Math">\iota_g</span> the inner automorphism of <span class="Math">G</span> that conjugates by <span class="Math">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="Math">X</span> and <span class="Math">Y</span> be topological spaces with universal covers <span class="Math">p \colon \tilde{X} \to X</span> and <span class="Math">q \colon \tilde{Y} \to Y</span> and let <span class="Math">\mathcal{D}(X), \mathcal{D}(Y)</span> be their covering transformations groups. Let <span class="Math">f,g \colon X \to Y</span> be continuous maps with lifts <span class="Math">\tilde{f}, \tilde{g} \colon \tilde{X} \to \tilde{Y}</span>. By <span class="Math">f_*\colon \mathcal{D}(X) \to \mathcal{D}(Y)</span>, denote the group homomorphism defined by <span class="Math">\tilde{f} \circ \gamma = f_*(\gamma) \circ \tilde{f}</span> for all <span class="Math">\gamma \in \mathcal{D}(X)</span>, and let <span class="Math">g_*</span> be defined similarly. The set of coincidence points <span class="Math">\operatorname{Coin}(f,g)</span> equals the union</p>

<p class="pcenter">\operatorname{Coin}(f,g) = \bigcup_{\alpha \in \mathcal{D}(Y)}
 p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g})).</p>

<p>For any two elements <span class="Math">\alpha, \beta \in \mathcal{D}(Y)</span>, the sets <span class="Math">p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g}))</span> and <span class="Math">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="Math">\gamma \in \mathcal{D}(X)</span> such that <span class="Math">\alpha = f_*(\gamma)^{-1} \circ \beta \circ g_*(\gamma)</span>, which is exactly the same as saying that <span class="Math">\alpha</span> and <span class="Math">\beta</span> are <span class="Math">(f_*,g_*)</span>-twisted conjugate. Thus,</p>

<p class="pcenter">\operatorname{Coin}(f,g) = \bigsqcup_{[\alpha]}
 p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g})),</p>

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

<p class="pcenter">R(f_*,g_*) \leq \left|\operatorname{Coin}(f,g)\right|,</p>

<p>whereas if <span class="Math">R(f_*,g_*) = \infty</span> there exist continuous maps <span class="Math">f'</span> and <span class="Math">g'</span> homotopic to <span class="Math">f</span> and <span class="Math">g</span> respectively such that <span class="Math">\operatorname{Coin}(f',g') = \varnothing</span>.</p>


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