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Philippe Tranchida

Publications and source records attributed to Philippe Tranchida.

16 recordsLinked to original sources
arXiv2026

Coset geometries acting on their elements: The $j$-diagonals twisting

Let $\beta$ be a coset incidence system and fix a type $j$. We use the orbits of the group of $\beta$ on pairs of distinct $j$-elements to define a Coxeter graph. The action on the $j$-elements induces an action on the corresponding Coxeter group, so the twisting construction for coset incidence systems can be applied. The resulting coset geometry, called the $j$-diagonals twisting, is always a regular hypertope if $\beta$ is a regular hypertope. Using this, we show that finite regular hypertope whose diagram is a tree with all but one of the labels equal to four always exist. We also define an extension operation that combines this Coxeter graph with another given Coxeter graph. For regular polytopes, these constructions recover the twisting extensions of McMullen and Schulte.

math.CO↗
arXiv2026

The geometry of wreath and semi-direct products

Coset geometries are incidence geometries constructed from a group $G$ and a system of subgroups $(G_i)_{i \in I}$ of subgroups of $G$. For any algebraic group operation, it is then natural to wonder whether it can be extended to the framework of coset geometries. This has been achieved in the case of the halving (\cite{halving}) and in the case of free (amalgamated) products, HNN-extensions, and semi-direct products (\cite{piedade2025group}). In this article, we explore more deeply two operations related to semi-direct products: the twisting and the wreath product. We show that these operations extend to coset geometries in such a way that they preserve key properties, such as flag-transitivity, residual-connectedness and being thin. In particular, we can apply twistings and wreath products to polytopes and hypertopes. Doing so, we show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group.

math.GR↗
arXiv2025

Groups represented by incidence geometries

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. Hence, in this setting, the group Inn(G) preserves the types of the associated incidence structure while the group Out(G) is acting non-trivially on the typeset of it, realizing the outer automorphisms of G as correlations. 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.

math.GR↗
arXiv2025

From Group Operations to Geometric Structures: Amalgamations, HNN-Extensions, and Twisting in Coset Geometries

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.

math.GR↗
arXiv2025

Geometries with trialities arising from linear spaces

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. Specifically, for any rank two incidence system $\Gamma$, we define its triangle complex $\Delta(\Gamma)$, a rank three incidence system whose elements consist of three copies of the flags (pairs of incident elements) of $\Gamma$. This triangle complex always admits a triality that cyclically permutes the three copies. We then explore in detail the properties of the triangle complex when $\Gamma$ is a linear space, including flag-transitivity, the existence of dualities, and connectivity properties. As a consequence of our work, this construction yields the first infinite family of thick, flag-transitive and residually connected geometries with trialities but no dualities.

math.CO↗
arXiv2024

Triples of involutions in PGL(2,q) and their incidence geometries

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 $\pi = PG(2,q)$. In that setting, involutions of $PGL(2,q)$ correspond bijectively to points of $\pi$ not in $O$. Triples of involutions $\{ \alpha_P,\alpha_Q,\alpha_R \}$ of $PGL(2,q)$ can then be seen also as triples of points $\{P,Q,R\}$ of $\pi$. We investigate the interplay between algebraic properties of the group $H = \langle \alpha_P,\alpha_Q,\alpha_R \rangle$ generated by three involutions and geometric properties of the triple of points $\{P,Q,R\}$. In particular, we show that the coset geometry $\Gamma = \Gamma(H,(H_0,H_1,H_2))$, where $H_0 = \langle \alpha_Q,\alpha_R \rangle, H_1 = \langle \alpha_P,\alpha_R \rangle$ and $H_2 = \langle \alpha_P,\alpha_Q \rangle$ is a regular hypertope if and only if $\{P,Q,R\}$ is a strongly non self-polar triangle, a terminology we introduce. 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 diagrams 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 $O$.

math.GR↗
arXiv2024

Constructing new geometries: a generalized approach to halving for hypertopes

Given a residually connected incidence geometry $\Gamma$ that satisfies two conditions, denoted $(B_1)$ and $(B_2)$, we construct a new geometry $H(\Gamma)$ with properties similar to those of $\Gamma$. This new geometry $H(\Gamma)$ is inspired by a construction of Percsy, Percsy and Leemans [1]. We show how $H(\Gamma)$ relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.

math.CO↗
arXiv2023

Flag transitive geometries with trialities and no dualities coming from Suzuki groups

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 field automorphisms. We also construct an infinite family of regular hypermaps with automorphism group Sz(q) that admit trialities but no dualities.

math.GR↗
arXiv2023

On trialities and their absolute geometries

We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type $(I_\sigma)$ and the absolute geometry is a generalized hexagon, the moving absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram with parameters $(d_p, g, d_L) = (5, 3, 6)$ for the groups $G_2(k)$ and $^3D_4(k)$, for any integer $k \geq 2$. We also classify the classical absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group $L_2(q^3)$, where $q$ is a power of a prime. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.

math.GR↗
arXiv2023

Thurston unit ball of a family of $n$-chained links and their fibered face

We determine the Thurston unit ball of a family of $n$-chained link, denoted by $C(n,p)$, where $n$ is the number of link components and $p$ is the number of twists. When $p$ is strictly positive, we prove that the Thurston unit ball for $C(n,p)$ is 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 any $p$. Finally we provide the Teichm\"uller polynomial for the face of Thurston unit ball of $C(n, -2)$ with $n\geq 3$.

math.GT↗
arXiv2022

Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology of mapping tori

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 entropy of a pseudo-Anosov map is bounded from above by $\dfrac{(k+1)\log(k+3)}{|\chi(S)|}$ 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.

math.GT↗
arXiv2022

Liftable automorphisms of right-angled Artin groups

Given a regular covering map $\varphi:\Lambda \to \Gamma$ of graphs, we investigate the subgroup $\operatorname{LAut}(\varphi)$ of the automorphism group $\operatorname{Aut}(A_\Gamma)$ of the right-angled Artin group $A_\Gamma$. This subgroup comprises all automorphisms that can be lifted to automorphisms of $A_\Lambda$. We first show that $\operatorname{LAut}(\varphi)$ is generated by a finite subset of Laurence's elementary automorphisms. For the subgroup $\operatorname{FAut}(\varphi)$ of $\operatorname{Aut}(A_\Lambda)$, which consists of lifts of automorphisms in $\operatorname{LAut}(\varphi)$, there exists a natural homomorphism $\operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi)$ induced by $\varphi$. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup $\operatorname{IA}(A_\Lambda)$ and deduce a short exact sequence reminiscent of results from the Birman--Hilden theory for surfaces.

math.GR↗
arXiv2020

Topological and dynamical properties of Torelli groups of partitioned surfaces

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, first 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 surfaces 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 first and second authors on Torelli groups for closed surfaces.

math.GT↗
arXiv2019

A new algorithm to classify chiral polytopes with a given automorphism group

We present a new algorithm to compute all the chiral polytopes that have a given group $G$ as full automorphism group. This algorithm uses a new set of generators that characterize the group, all of them except one being involutions. It permits to compute all chiral polytopes of groups that were previously unreachable by other known algorithms.

math.GR↗