Group actions on contractible $2$-complexes II

In this second part we prove the following:

This completes the proof of the following result: every action of a finite group $G$ on a finite and contractible $2$-complex $X$ has a fixed point.

The groups $G$ considered in [SC21, Theorem B] share a key property: they admit a nontrivial representation ${\rho_{0}}$ which restricts to an irreducible representation on the Borel subgroup. The moduli $\mathcal{M}_{k}$ of representations of $\Gamma_{k}=X_{1}^{OS+k}(G):G$ constructed in the proof of [SC21, Theorem B] is built from a representation with this property. When $q\equiv 1\pmod{4}$ no nontrivial representation of $\mathrm{PSL}_{2}(q)$ restricts to an irreducible representation on the Borel subgroup. To prove Theorem C, we circumvent this difficulty by instead considering the action of $\widehat{G}=\mathrm{SL}_{2}(q)$ on $X_{1}^{OS+k}(G)$.

Acknowledgements. This work was partially done during a stay of the first author at The Mathematisches Forschungsinstitut Oberwolfach. He is very grateful to the MFO for their hospitality and support.

We denote the set of eigenvalues of a square matrix $M$ by $\Lambda(M)$.

Let $G=\mathrm{PSL}_{2}(q)$ with $q\equiv 5\pmod{24}$ or $q\equiv 13\pmod{24}$. Let $\widehat{G}=\mathrm{SL}_{2}(q)$, so that ${\mathbf{Z}}(\widehat{G})=\{1,-1\}$ and $\widehat{G}/{\mathbf{Z}}(\widehat{G})=G$. In what follows we denote $z=-1\in{\mathbf{Z}}(\widehat{G})$.

We consider a construction of $X_{1}^{OS}(G)$ as in [SC21, Proposition 5.5]. Recall that for any $k\geq 0$, we can also consider the $G$-graph $X_{1}^{OS+k}(G)$ obtained from $X_{1}^{OS}(G)$ by attaching $k$ free orbits of $1$-cells (note that by [SC20, Proposition 3.10] the $G$-homotopy type of $X_{1}^{OS+k}(G)$ does not depend on the particular way these free orbits are attached).

We consider the action of $\widehat{G}=\mathrm{SL}_{2}(q)$ on $X_{1}^{OS+k}(G)$ defined using the projection $\pi\colon\widehat{G}\to G$. The stabilizer of a vertex (resp. edge) for the action of $\widehat{G}$ is a central extension, by ${\mathbf{Z}}(\widehat{G})$, of the stabilizer for the action of $G$. Then the $\widehat{G}$-orbits are connected as in Figure 1. The group $B$ denotes the Borel subgroup of $\widehat{G}$ and $Q_{8}$ denotes the quaternion group.

We now apply Brown’s result [SC21, Theorem 3.1]. The choices in each case are the following. Note that in each case the stabilizers are given in Tables 1 and 2.

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  For $\widehat{G}=\mathrm{SL}_{2}(q)$ with $q\equiv 13\pmod{24}$ we take $V=\{v_{0},v_{1},v_{2},v_{3}\}$, $E=\{\eta_{0},\eta_{1},\eta_{2},\allowbreak\eta_{3},\allowbreak\eta^{\prime}_{1},\ldots,\allowbreak\eta^{\prime}_{k}\}$, $T=\{\eta_{0},\eta_{1},\eta_{2}\}$, with $v_{0}\xrightarrow{\eta_{0}}v_{1}$, $v_{1}\xrightarrow{\eta_{1}}v_{2}$, $v_{1}\xrightarrow{\eta_{2}}v_{3}$, $v_{3}\xrightarrow{\eta_{3}}g_{\eta_{3}}v_{0}$ and $v_{0}\xrightarrow{\eta^{\prime}_{i}}v_{0}$ for $i=1,\ldots,k$.

• •

  For $\widehat{G}=\mathrm{SL}_{2}(q)$ with $q\equiv 5\pmod{24}$ we take $V=\{v_{0},v_{1},v_{2},v_{3}\}$, $E=\{\eta_{0},\eta_{1},\eta_{2},\allowbreak\eta_{3},\allowbreak\eta^{\prime}_{1},\ldots,\eta^{\prime}_{k}\}$, $T=\{\eta_{0},\eta_{2},\eta_{3}\}$, with $v_{0}\xrightarrow{\eta_{0}}v_{1}$, $v_{1}\xrightarrow{\eta_{2}}v_{3}$, $v_{3}\xrightarrow{\eta_{3}}v_{2}$, $v_{2}\xrightarrow{\eta_{1}}g_{\eta_{1}}v_{0}$ and $v_{0}\xrightarrow{\eta^{\prime}_{i}}v_{0}$ for $i=1,\ldots,k$.

In what follows $\Gamma_{k}$ is the group obtained by applying Brown’s result to the action of $G$ on $X_{1}^{OS+k}(G)$ with these choices. The following lemma is an extension of [SC21, Lemma 5.6] for the action of $\widehat{G}$ on $X_{1}^{OS}(G)$.

In this section we fix a suitable irreducible representation ${\rho_{0}}\colon\widehat{G}\to\mathbb{G}$ and compute the dimension of the centralizers ${\mathbf{C}}_{\mathbb{G}}({\rho_{0}}(\widehat{G}_{\eta_{i}}))$ and ${\mathbf{C}}_{\mathbb{G}}({\rho_{0}}(\widehat{G}_{v_{i}}))$. To perform these computations, we first need to know how many elements of each conjugacy class of $\widehat{G}$ appear in each of the subgroups $\widehat{G}_{\eta_{i}}$ and $\widehat{G}_{v_{i}}$. Recall that if $x$ is an element of $\widehat{G}$, then $(x)$ denotes its conjugacy class. For the structure and conjugacy classes of the groups $\mathrm{SL}_{2}(q)$ we refer to [SC21, Appendix A.5].

For $i=1,3$ and each of the groups $G$ in Theorem C, we fix a generator $\hat{g}_{i}$ of $\widehat{G}_{\eta_{i}}$.

For each of the groups $G$ in Theorem C, we consider a closed edge path $\xi$ in $X_{1}^{OS}(G)$ such that attaching a free orbit of $2$-cells along this path gives an acyclic $2$-complex. We define $x_{0}=i(\xi)$, where $i\colon\pi_{1}(X_{1}^{OS}(G),v_{0})\to\Gamma_{0}$ is the inclusion given by Brown’s theorem. We set $x_{i}=x_{\eta^{\prime}_{i}}$ for $i=1,\ldots,k$. Let ${\tilde{\eta}}$ be the unique edge of $X_{1}^{OS}(G)$ which lies in $E-T$. We define $y_{0}=x_{{\tilde{\eta}}}$ and $y_{i}=x_{\eta^{\prime}_{i}}$ for $i=1,\ldots,k$.

Let $\mathcal{M}_{k}$ be the moduli of representations of $\Gamma_{k}$ obtained from the representation ${\rho_{0}}\colon\widehat{G}\to\mathbf{U}(m)$ of Proposition 4.0.2 using [SC21, Theorem 3.2]. Let $\overline{\mathcal{M}}_{k}$ be the corresponding quotient obtained using [SC21, Proposition 3.3]. Note that the equalities $\mathcal{M}_{k}=\mathcal{M}_{0}\times\mathbb{G}^{k}$ and $\overline{\mathcal{M}}_{k}=\overline{\mathcal{M}}_{0}\times\mathbb{G}^{k}$ still hold, because ${\rho_{0}}(z)\in{\mathbf{Z}}(\mathbb{G})$.

In what follows we consider the induced maps $X_{i}(\tau)=\rho_{\tau}(x_{i})$, $Y_{i}(\tau)=\rho_{\tau}(y_{i})$.

We now prove the degree of $\overline{Y}_{0}$ is $0$ for each of the groups in Theorem C. When $q\equiv 13\pmod{24}$, the approach is similar to that of [SC21, Propositions 9.1 and 9.2]. When $q\equiv 5\pmod{24}$ the approach needs to be modified. Table 3 gives the value of $Y_{0}$ in the different cases that we consider.