Abstract We construct novel families of exact immersed and embedded Lagrangian translating solitons and special Lagrangian submanifolds in $$\mathbb {C}^m$$ C m that are invariant under the action of various admissible compact subgroups $$G \le {{\,\textrm{SU}\,}}(m-1)$$ G ≤ SU ( m - 1 ) with cohomogeneity-two. These examples are obtained via an Ansatz generalising a construction of Castro–Lerma in $$\mathbb {C}^2$$ C 2 . We give explicit examples of admissible group actions, including a full classification for G simple. We also describe novel Lagrangian translators symmetric with respect to non-compact subgroups of the affine special unitary group $${{\,\textrm{SU}\,}}(m)\ltimes \mathbb {C}^m$$ SU ( m ) ⋉ C m , including cohomogeneity-one examples.

Abstract In this paper, we show that wreath products of groups have linear divergence, and we generalise the argument to permutational wreath products. We also prove that Houghton groups $${\mathcal {H}}_m$$ H m with $$m\ge 2$$ m ≥ 2 and Baumslag-Solitar groups have linear divergence. We explain how to generalise the argument for wreath products so that it holds for halo products of groups whose halo is large-scale commutative. Finally, we show that wreath products of graphs and Diestel-Leader graphs have linear divergence. The argument for Diestel-Leader graphs is further generalised to horocyclic products of proper, geodesically complete, Busemann $$\delta $$ δ -hyperbolic spaces that are uniformly not a quasi-line.

Abstract We study projective deformations of (topologically finite) hyperbolic 3-orbifolds whose ends have turnover cross section. These deformations are examples of projective cusp openings , meaning that hyperbolic cusps are deformed in the projective setting such that they become totally geodesic generalized cusps with diagonal holonomy. We find that this kind of structure is the only one that can arise when deforming hyperbolic turnover cusps, and that turnover funnels remain totally geodesic. Therefore, we argue that, under no infinitesimal rigidity assumptions, the deformed projective 3-orbifold remains properly convex. Additionally, we give a complete description of the character variety $$X(\pi _1S^2(3,3,3),\mathrm {SL_4\mathbb {R}})$$ X ( π 1 S 2 ( 3 , 3 , 3 ) , SL 4 R ) .

Abstract In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi–homogeneous singularities. First, we introduce a new tool that enables us to characterize the property of being plus–one generated within the class of conic arrangements with some naturally chosen quasi–homogeneous singularities. Next, we present a classification result on plus–one generated conic arrangements admitting only nodes and tacnodes as singularities. Building on results regarding conic arrangements with nodes and tacnodes, we present new examples of strong Ziegler pairs of conic-line arrangements – that is, arrangements having the same strong combinatorics but distinct derivation modules.

Abstract We deal with generalizations of the Fundamental Theorem of Projective Geometry to other related geometries (of dimension $$\ge 3$$ ≥ 3 ) and non bijective maps. We consider locally projective geometries and locally affino-projective geometries (which include the classical Möbius, Laguerre and Minkowski geometries).