Let Γ be a non-Euclidean crystallographic group. Γ is said to be non-maximal if there exists a non-Euclidean crystallographic group Γ such that Γ Γ and the dimension of the Teichmüller space of Γ equals the dimension of the Teichmüller space of Γ . The full list of such pairs of groups is computed in the case when Γ is non-normal in Γ . The corresponding problem for Fuchsian groups was solved by Singerman.

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Gossip is a pervasive phenomenon in organizations causing many individuals to have second-hand information about their colleagues. However, whether it is used to inform friendship choices (i.e., friendship creation, friendship maintenance, friendship discontinuation) is not that evident. This paper articulates and empirically tests a complex contagion model to explain how gossip, through its reputational effects, can affect the evolution of friendship ties. We argue that hearing gossip from more than a single sender (and about several targets) impacts receivers’ friendships with the gossip targets. Hypotheses are tested in a two-wave sociometric panel study among 148 employees in a Dutch childcare organization. Stochastic actor-oriented models reveal positive gossip favors receiver-target friendships, whereas negative gossip inhibits them. We also find evidence supporting that, for damaging relationships, negative gossip needs to originate in more than a single sender. Positive gossip about a high number of targets discourages friendships with colleagues in general, while negative gossip about many targets produces diverging trends. Overall, the study demonstrates that second-hand information influences the evolution of expressive relations. It also underscores the need to refine and extend current theorizing concerning the multiple (and potentially competing) psychological mechanisms causing some of the observed effects.