By Karel Dekimpe, Paul Igodt, Alain Valette

This quantity stories on study concerning Discrete teams and Geometric constructions, as provided through the foreign Workshop held could 26-30, 2008, in Kortrijk, Belgium. Readers will take advantage of amazing survey papers by way of John R. Parker on ways to build and research lattices in complicated hyperbolic house and by way of Ursula Hamenstadt on houses of workforce activities with a rank-one aspect on right CAT (0)-spaces. This quantity additionally includes learn papers within the sector of team activities and geometric constructions, together with paintings on loops on a two times punctured torus, the simplicial quantity of goods and fiber bundles, the homology of Hantzsche - Wendt teams, tension of actual Bott towers, circles in teams of tender circle homeomorphisms, and teams generated via backbone reflections admitting crooked basic domain names

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18 of [12]). 4). 2 of [57]). Suppose that m ∈ {5, 6, 7, 8, 9, 10, 12, 18}. Let Γ be a Deligne-Mostow group of type p = m, k = 2 generated by R1 , A1 and J. Write R2 = JR1 J −1 and R3 = J −1 R1 J. Then the group Γ∗ generated by ∗ R1∗ = R2−1 R1 R2 , R 1 = R3 , J ∗ = J −1 is a Deligne-Mostow group of type p = m, k = m/2. Moreover Γ∗ is isomorphic to Γ. The following table gives a summary of the list of 46 Deligne-Mostow lattices with three fold symmetry. Of these groups, 41 satisfy the orbifold condition and the remaining 5 are related to a group satisfying the orbifold condition by a commensurability theorem (the latter are the groups in the following table for which d is not an integer).

The stabiliser of z134 is the group A1 , R2 . This group has order 8p2 k2 /(2p + 2k − pk)2 = 2l2 . It is a central extension of the orientation preserving subgroup of a (2, p, k) triangle group (which has order 4pk/(2p + 2k − pk) = −2l) by a cyclic group of order 2pk/(2p + 2k − pk) = −l. Proof. 4. In this case we lift z134 to a vector z134 which spans U . This is a common eigenspace of R2 and A1 . Once again we list their eigenvalues, with the eigenvalue corresponding to U third. • R2 has eigenvalues e4πi/3p , e−2πi/3p , e−2πi/3p , • A1 has eigenvalues e4πi/3k , e−2πi/3k , e−2πi/3k , • R2 A1 has eigenvalues ieπi/3p+πi/3k , −ieπi/3p+πi/3k , e−2πi/3p−2πi/3k .

The stabiliser of z134 is the group A1 , R2 . This group has order 8p2 k2 /(2p + 2k − pk)2 = 2l2 . It is a central extension of the orientation preserving subgroup of a (2, p, k) triangle group (which has order 4pk/(2p + 2k − pk) = −2l) by a cyclic group of order 2pk/(2p + 2k − pk) = −l. Proof. 4. In this case we lift z134 to a vector z134 which spans U . This is a common eigenspace of R2 and A1 . Once again we list their eigenvalues, with the eigenvalue corresponding to U third. • R2 has eigenvalues e4πi/3p , e−2πi/3p , e−2πi/3p , • A1 has eigenvalues e4πi/3k , e−2πi/3k , e−2πi/3k , • R2 A1 has eigenvalues ieπi/3p+πi/3k , −ieπi/3p+πi/3k , e−2πi/3p−2πi/3k .