UNIMODULAR GROUPS OF TYPE ℝ³⋊ ℝ

Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups

“…Note that M is isomorphic to the solvable Lie group G 6 (1) in [27, p. 98]. The Lie group M has no lattices [55].…”

Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures

“…[30]), and therefore Sol 3 × R admits lattices as well. Now we use the classification given in [28] Finally, we take into account the non-completely solvable Lie algebras r 3,0 × R and r 4,λ,−λ/2 (λ > 0). It is well known that the simply connected Lie group E(2) corresponding to r 3,0 e(2) admits lattices (this fact is also an easy application of Proposition 2.3), and therefore E(2) × R admits lattices, too.…”

“…In this section we will determine up to Lie algebra isomorphism the almost abelian Lie algebras in Theorem 4.5 whose associated simply connected Lie groups admit lattices. In order to do this we use the classification of unimodular completely solvable Lie groups of type R R 3 in [28] and Proposition 2.3. Theorem 4.8.…”

“…It is well known that the simply connected Lie group Sol 3 corresponding to r 3,−1 sol 3 admits lattices (see for instance [30]), and therefore Sol 3 × R admits lattices as well. Now we use the classification given in [28], noting that r 4,λ,−(1+λ) sol 0 = G 0 from §3.2.2. Finally, we take into account the non-completely solvable Lie algebras r 3,0 × R and r 4,λ,−λ/2 (λ > 0).…”

Structure of locally conformally symplectic Lie algebras and solvmanifolds

“…12 a 32`a13 a 33 , a 13 a 32´a12 a 33 , a 24 a 32´a33 a 34 , a 2 24 , a 12 γ`a 13 , a 13 γ´a 12 , a 24 γ´a 34 , a 32 γ`a 33 , a 33 γ´a 32 , a 34 γ`a 24 , a 41 γ`1 2 , a 11 , a 21 , a 22´a33 , a 23`a32 , a31 .ẇ hich gives that A has determinant 0.…”