Endomorphisms of mapping tori (Q6571603)

The author analyzes algebraic properties of mapping tori of Poincaré Duality groups and, using them, he derives geometric results. The main technical assumption on a group \(K\) is that, if a power of an element \(\theta \in Out(K)\) is conjugate to itself, then \(\theta\) is a torsion element. One of the main results of the paper is the following. Let \(E_h\) be the mapping torus of a homeomorphism \(h\) of a closed aspherical manifold \(F\), which has non-zero Euler characteristic and residually finite fundamental group. If \(\pi_1(F)\) satisfies the above condition, then the following are equivalent:\N\begin{itemize}\N\item[1.] \(\pi_1(E_h)\) has trivial center.\N\item[2.] Every endomorphism of \(\pi_1(E_h)\) onto a finite index subgroup induces a homotopy equivalence on \(E_h\).\N\item[3.] Every injective endomorphism of \(\pi_1(E_h)\) induces a homotopy equivalence on \(E_h\).\N\item[4.] Every self-map of \(E_h\) of non-zero degree is a homotopy equivalence.\N\end{itemize}\NIf, moreover, \(F\) is topologically rigid, then homotopy equivalence in (2), (3) and (4) can be replaced by a map homotopic to a homeomorphism.\N\NThe author remarks that his approach is purely algebraic and the group does not have to be the fundamental group of an aspherical manifold, he just used the algebraic properties of such a group. More specifically:\N\NThe algebraic result that is quite useful in the constructions is the following. Let \(K\) be a residually finite group of type FP, \(cd(K) = n\), such that \({\chi}(K) \not= 0\), \(H^n(K; \mathbb{R}) \not= 0\) and \(H^n(K; \mathbb{Z}K)\) is finitely generated. If \(K\) satisfies the basic condition, then the following are equivalent for the mapping torus \(\Gamma_{\theta} = K \times_{\theta} \mathbb{Z}\) of any automorphism \(\theta\) of \(K\):\N\begin{itemize}\N\item[1.] \(C(\Gamma_{\theta}) = 1\).\N\item[2.] \(\Gamma_{\theta}\) is cofinitely Hopfian.\N\item[3.] \(\Gamma_{\theta}\) is co-Hopfian.\N\end{itemize}\NFor applications, the first result is that the mapping torus \(E_h\) of a self-diffeomorphism of a closed hyperbolic surface admits a self-map of degree greater than one if and only if \(h\) is periodic. Also, for \(E_h\) as in the first result above with \(F\) topologically rigid, then every self-map of \(E_h\) is either homotopic to a homeomorphism or to a non-trivial finite cover. For \(E_h\) as above, there is a functorial semi-norm that is non-zero and finite on \(E_h\) if and only if \(h_*\) has infinite order in \(Out(\pi_1(F))\). Another application is on Gromov's rigidity and domination. Let \(E_{h_i}\), \(i = 1, 2\), as in the first result above and \(C({\pi}_1(E_{h_i})) = 1\). If \(E_{h_1}\) and \(E_{h_2}\) dominate each other then they are homotopy equivalent. If the \(F_i\) are topologically rigid then they are homeomorphic. If \(M\) is a manifold whose fundamental group satisfies the algebraic conditions above and has trivial center, then any quasiregular map \(f: M \to M\) is a homeomorphism. The next result connects Euler characteristics, Lefschetz numbers and \(L^2\)-Betti numbers of groups. Let \(K\) be a group, \(cd(K) < \infty\) and \({\chi}(K) \not= 0\). Assume that some power \(d > 1\) of \(\theta \in Out(K)\) is conjugate to \(\theta\). Then there is an integer \(q_m = ({\chi}(K), d, m)\) such that \({\chi}(K) = \Lambda(\theta^{q_m})\), where \(\Lambda\) denotes the Lefschetz number.