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Algebraic Combinatorics
In the study of polytopes, there is a correspondence between the most symmetric (regular) polytopes and a class of groups called string C-groups. There is a similar correspondence between chiral polytopes and string C+-groups. The theory of hypertopes (regular and chiral) studies a broader class of groups called C-groups and C+-groups. From a C-group G, an incidence geometry X can be constructed. It is then possible to check whether X is residually connected directly from algebraic properties of G. In this article, we show that it is also possible to do so if we start from a C+-group.
Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry
Based on Leeman's previous work, we write a concrete implementation of an algorithm that finds all chiral polytopes having a given automorphism group. This is not the first algorithm to do so, but this one is significantly faster, as we show numerically in the table at the end of the article. Thanks to this new algorithm, we find new interesting examples of chiral polytopes.
Geometriae Dedicata
A. Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned sur-faces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the rst and second authors on Torelli groups for closed surfaces.
Bulletin of the Australian Mathematical Society
We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least three vertices are not relatively hyperbolic. We then show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, the outer automorphism groups are virtually isomorphic to either a finite group, an infinite cyclic group or GL2(Z).
Glasnik Matematicki
We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface S of genus g with n punctures, we show that the minimal entropy of a pseudo-Anosov map is bounded from above by [(k + 1) log(k + 3) / |χ(S)|] up to a constant multiple when the rank of the first homology of the mapping torus is k + 1 and k, g, n satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.
Advances in geometry
Given an incidence geometry X together with a triality T, or any correlation in fact, we can construct the absolute geometry of X with respect to T, a classical and well-studied geometry. This geometry is made of the fixed elements by T of X. In some cases, in particular if X is thin, it turns out that the absolute geometry is poorly connected. We introduce a different version of an absolute geometry, that we call the moving absolute geometry, as it also considers some elements that are moved by T. Absolute geometries were used to construct generalized hexagons by Tits in 1959. We show that in the case investigated by Tits, the moving absolute geometries are highly symmetric rank 2 incidence geometries with Buekenhout diagram (5,3,6). We then investigate the absolute geometries of the geometries with trialities but no dualities constructed by Leemans and Stokes.
Journal of Group Theory
We investigate the relation between covering of graphs and the automorphisms group of the associated Right-angled Artin-Tits groups. We define a notion of liftable automorphism and show that the group of liftable automorphisms is finitely generated. We then show that the group of lifts of the identity is commensurable to the Torelli subgroup of the RAAG of the covering graph. This implies the existence of a Birman-Hilden sort of exact sequence.
Osaka Journal of Mathematics
We determine the Thurston unit ball of a family of n-chained link with p half-twists on one component. This links are denoted by C(n; p). When p is strictly positive, we prove that the Thurston unit ball for C(n; p) contains an n-dimensional cocube, for arbitrary n. Moreover, we clarify the condition for which C(n; p) is fibered and find at least one fibered face for each case. Finally, we provide the Teichmuller polynomial for a face of the Thurston unit ball of C(n;-2) for arbitrary n 3.
Flag transitive geometries with trialities but no dualities from Suzuki groups (Leemans, Stokes, T.)
Flag transitive geometries with trialities but no dualities from Suzuki groups (Leemans, Stokes, T.)
Journal of Combinatorial Theory, Series A
Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups PSL(2, q) (where q = p^(3n) with p a prime and n > 0 a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we construct the first infinite family of incidence geometries of rank three that are flag transitive and have trialities but no dualities. These geometries are constructed using chamber systems of Suzuki groups Sz(q) (where q = 2^(2e+1) with e a positive integer and 2e+1 is divisible by 3) and the trialities come from eld automorphisms. We also construct an infinite family of regular hypermaps with automorphism group Sz(q) that admit trialities but no dualities.
Innovation in Incidence Geometry
For q = p^n with p an odd prime, the projective linear group PGL(2, q) can be seen as the stabilizer of a conic O in a projective plane π = PG(2, q). In that setting, involutions of PGL(2, q) correspond bijectively to points of π not in O. Triples of involutions {α_P , α_Q, α_R} of PGL(2, q) correspond in this way to triple of points {P, Q,R} of π. We investigate the interplay between algebraic properties of the group H = ⟨α_P , α_Q, α_R⟩ and geometric properties of the triple {P, Q,R}. In particular, we show that the coset geometry Γ = Γ(H, (H0,H1,H2)), where H0 = ⟨αQ, αR⟩,H1 = ⟨αP , αR⟩ and H2 = ⟨αP , αQ⟩ is a regular hypertope if and only if {P, Q,R} is a strongly non self-polar triangle. This entirely characterizes hypertopes of rank 3 with automorphism group a subgroup of PGL(2, q). As a corollary, we obtain the existence of hypertopes of rank 3 with non linear diagram and with automorphism group PGL(2, q), for any q = p^n with p an odd prime. We also study in more details the case where the triangle {P, Q,R} is tangent to O.
Combinatorica
Halving is a classical operation on abstract polytopes. We show how this operations can be extended to hypertopes that have a diagrams admitting a non-degenerate leaf. The definition of the operation is purely algebraic, but we show how to interpret the result geometrically. Indeed, we show that the halved-group is the automorphism group of an hypertope, whic can be constructed from the starting hypertope by coloring or doubling the elements of various types.
A triality is a sort of super-symmetry that exchanges the types of the elements of an incidence geometry in cycles of length three. Although geometries with trialities exhibit fascinating behaviors, their construction is challenging, making them rare in the literature. To understand trialities more deeply, it is crucial to have a wide variety of examples at hand. In this article, we introduce a general method for constructing various rank-three incidence systems with trialities.
Coset incidence geometries, introduced by Jacques Tits, provide a versatile framework for studying the interplay between group theory and geometry.
In this article, we build upon that idea by extending classical group-theoretic constructions (amalgamated products, HNN-extensions, semi-direct products, and twisting) to the setting of coset geometries. This gives a general way to glue together incidence geometries in various ways. This provides a general framework for combining or gluing incidence geometries in different ways while preserving essential properties such as flag-transitivity and residual connectedness.
Using these techniques, we analyze families of Shephard groups, which generalize both Coxeter and Artin-Tits groups, and their associated simplicial complexes.Our results also point to the existence of a Bass-Serre theory for coset geometries and of a fundamental geometry of a graph of coset geometries.
Preprint
The aim of this paper is to use the framework of incidence geometry to develop a theory that permits to model both the inner and outer automorphisms of a group G simultaneously. More precisely, to any group G, we attempt to associate an incidence system whose group of type-preserving automorphisms is Inn(G), the group of inner automorphisms of G, and whose full group of automorphisms is the group Out(G) of outer automorphisms of G, getting what we call an incidence geometric representation theory for groups. We give examples of incidence geometric representations for the dihedral groups, the symmetric groups, the automorphism groups of the five platonic solids, families of classical groups defined over fields such as the projective linear groups, and finally subgroups of free groups whose outer automorphism group is the largest finite subgroup of their automorphism group.
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