  
  [1X2 [33X[0;0YMathematical background[133X[101X
  
  [33X[0;0YLet [23XG[123X and [23XH[123X be groups and let [23X\varphi[123X and [23X\psi[123X be group homomorphisms from [23XH[123X
  to [23XG[123X. The pair [23X(\varphi,\psi)[123X induces a (right) group action of [23XH[123X on [23XG[123X given
  by the map[133X
  
  
  [24X[33X[0;6YG \times H \to G \colon (g,h) \mapsto \varphi(h)^{-1} g\,\psi(h).[133X
  
  [124X
  
  [33X[0;0YThis  group  action is called [13X[23X(\varphi,\psi)[123X-twisted conjugation[113X. The orbits
  are called [13XReidemeister classes[113X or [13Xtwisted conjugacy classes[113X, and the number
  of Reidemeister classes is called the [13XReidemeister number[113X [23XR(\varphi,\psi)[123X of
  the  pair  [23X(\varphi,\psi)[123X.  The  stabiliser  of  the  identity [23X1_G[123X under the
  [23X(\varphi,\psi)[123X-twisted  conjugacy  action  of  [23XH[123X  is exactly the [13Xcoincidence
  group[113X[133X
  
  
  [24X[33X[0;6Y\operatorname{Coin}(\varphi,\psi)  =  \left\{\,  h  \in  H \mid \varphi(h) =
  \psi(h) \, \right\}.[133X
  
  [124X
  
  [33X[0;0YGeneralising  this,  the  stabiliser of any [23Xg \in G[123X is the coincidence group
  [23X\operatorname{Coin}(\iota_g\varphi,\psi)[123X,    with    [23X\iota_g[123X    the    inner
  automorphism of [23XG[123X that conjugates by [23Xg[123X.[133X
  
  [33X[0;0YTwisted   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 [23XX[123X and [23XY[123X be topological spaces with universal covers
  [23Xp   \colon   \tilde{X}   \to  X[123X  and  [23Xq  \colon  \tilde{Y}  \to  Y[123X  and  let
  [23X\mathcal{D}(X), \mathcal{D}(Y)[123X be their covering transformations groups. Let
  [23Xf,g \colon X \to Y[123X be continuous maps with lifts [23X\tilde{f}, \tilde{g} \colon
  \tilde{X}  \to  \tilde{Y}[123X.  By  [23Xf_*\colon \mathcal{D}(X) \to \mathcal{D}(Y)[123X,
  denote   the   group  homomorphism  defined  by  [23X\tilde{f}  \circ  \gamma  =
  f_*(\gamma)  \circ  \tilde{f}[123X for all [23X\gamma \in \mathcal{D}(X)[123X, and let [23Xg_*[123X
  be defined similarly. The set of coincidence points [23X\operatorname{Coin}(f,g)[123X
  equals the union[133X
  
  
  [24X[33X[0;6Y\operatorname{Coin}(f,g)     =     \bigcup_{\alpha    \in    \mathcal{D}(Y)}
  p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g})).[133X
  
  [124X
  
  [33X[0;0YFor   any   two   elements   [23X\alpha,  \beta  \in  \mathcal{D}(Y)[123X,  the  sets
  [23Xp(\operatorname{Coin}(\tilde{f},        \alpha        \tilde{g}))[123X        and
  [23Xp(\operatorname{Coin}(\tilde{f},  \beta  \tilde{g}))[123X  are either disjoint or
  equal.  Moreover, they are equal if and only if there exists some [23X\gamma \in
  \mathcal{D}(X)[123X  such  that  [23X\alpha  =  f_*(\gamma)^{-1}  \circ  \beta  \circ
  g_*(\gamma)[123X,  which  is exactly the same as saying that [23X\alpha[123X and [23X\beta[123X are
  [23X(f_*,g_*)[123X-twisted conjugate. Thus,[133X
  
  
  [24X[33X[0;6Y\operatorname{Coin}(f,g)                =               \bigsqcup_{[\alpha]}
  p(\operatorname{Coin}(\tilde{f}, \alpha \tilde{g})),[133X
  
  [124X
  
  [33X[0;0Ywhere  [23X[\alpha][123X  runs  over  the  [23X(f_*,g_*)[123X-twisted  conjugacy  classes. For
  sufficiently  well-behaved  spaces  [23XX[123X  and  [23XY[123X  (e.g.  nilmanifolds  of equal
  dimension) we have that if [23XR(f_*,g_*) < \infty[123X, then[133X
  
  
  [24X[33X[0;6YR(f_*,g_*) \leq \left|\operatorname{Coin}(f,g)\right|,[133X
  
  [124X
  
  [33X[0;0Ywhereas  if  [23XR(f_*,g_*)  =  \infty[123X  there  exist  continuous  maps [23Xf'[123X and [23Xg'[123X
  homotopic  to  [23Xf[123X  and  [23Xg[123X respectively such that [23X\operatorname{Coin}(f',g') =
  \varnothing[123X.[133X
  
