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Javier Aramayona

Publications and source records attributed to Javier Aramayona.

At least 37 records · Page 2Linked to original sources
arXiv2017

Representations of pure symmetric automorphism groups of RAAGs

We study representations of the pure symmetric automorphism group $PAut(A_Γ)$ of a RAAG $A_Γ$ with defining graph $Γ$. We first construct a homomorphism from $PAut(A_Γ)$ to the direct product of a RAAG and a finite direct product of copies of $F_2 \times F_2$; moreover, the image of $PAut(A_Γ)$ under this homomorphism is surjective onto each factor. As a consequence, we obtain interesting actions of $PAut(A_Γ)$ on non-positively curved spaces We then exhibit, for connected $Γ$, a RAAG which property contains $Inn(A_Γ)$ and embeds as a normal subgroup of $PAut(A_Γ)$. We end with a discussion of the linearity problem for $PAut(A_Γ)$.

math.GR↗
arXiv2017

Trees, homology, and automorphism groups of RAAGs

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this subgroup, based on the number and degree of a certain type of vertices, which we call deep. We then use combinatorial methods to analyze the average value of this Betti number, in terms of the size of the defining tree.

math.GR↗
arXiv2017

On the first cohomology of automorphism groups of graph groups

We study the (virtual) indicability of the automorphism group $Aut(A_Γ)$ of the right-angled Artin group $A_Γ$ associated to a simplicial graph $Γ$. First, we identify two conditions -- denoted (B1) and (B2) -- on $Γ$ which together imply that $H^1(G, Z)=0$ for certain finite-index subgroups $G<Aut(A_Γ)$. On the other hand we will show that (B2) is equivalent to the matrix group ${\mathcal H} = {\rm Im}(Aut(A_Γ) \to Aut(H_1(A_Γ))) <GL(n,Z)$ not being virtually indicable, and also to $\mathcal H$ having Kazhdan's property (T). As a consequence, $Aut(A_Γ)$ virtually surjects onto $Z$ whenever $Γ$ does not satisfy (B2). In addition, we give an extra property of $Γ$ ensuring that $Aut(A_Γ)$ and $Out(A_Γ)$ virtually surject onto $Z$. Finally, in the appendix we offer some remarks on the linearity problem for $Aut(A_Γ)$.

math.GR↗
arXiv2015

Arc and curve graphs for infinite-type surfaces

We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured surfaces. Second, we study the subgraph of the curve graph spanned by those elements which intersect a fixed separating curve on the surface. We show that this graph has infinite diameter and geometric rank 3, and thus is not hyperbolic.

math.GT↗
arXiv2014

Injective maps between flip graphs

We prove that every injective simplicial map $\mathcal{F}(S) \to \mathcal{F}(S')$ between flip graphs is induced by a subsurface inclusion $S\to S'$, except in finitely many cases. This extends a result of Korkmaz--Papadopoulos which asserts that every automorphism of the flip graph of a surface without boundary is induced by a surface homeomorphism.

math.GT↗
arXiv2012

Finite rigid sets in curve complexes

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map from X into C(S) is the restriction of an element of Aut(C(S)), unique up to the (finite) point-wise stabilizer of X in Aut(C(S)). Furthermore, if S is not a twice-punctured torus, then we can replace Aut(C(S)) in this statement with the extended mapping class group.

math.GT↗
arXiv2011

Convexity of strata in diagonal pants graphs of surfaces

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

math.GT↗
arXiv2010

Homomorphisms between mapping class groups

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em embedding}, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the same hypotheses on genus, if $X$ and $Y$ have finite analytic type then every non-constant holomorphic map $\CM(X)\to\CM(Y)$ between the corresponding moduli spaces is a forgetful map. In particular, there are no such holomorphic maps unless $X$ and $Y$ have the same genus and $Y$ has at most as many marked points as $X$.

math.GT↗
arXiv2009

Simplicial embeddings between pants graphs

We prove that, except in some low-complexity cases, every locally injective simplicial map between pants graphs is induced by a $π_1$-injective embedding between the corresponding surfaces.

math.GT↗
arXiv2008

Injections of mapping class groups

We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary.

math.GT↗
arXiv2007

Constructing convex planes in the pants complex

Our main theorem identifies a class of totally geodesic subgraphs of the 1-skeleton of the pants complex, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the 1-skeleton of the pants complex.

math.GT↗
arXiv2006

Free subgroups of surface mapping class groups

We quantify the generation of free subgroups of surface mapping class groups by pseudo-Anosov mapping classes in terms of their translation distance and the distance between their axes. Our methods make reference to \teichmuller space only.

math.GR↗
arXiv2005

A simple criterion for non-relative hyperbolicity and one-endedness of groups

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively hyperbolic group structure and have one end. Applications include surface mapping class groups, the Torelli group, (special) automorphism and outer automorphism groups of most free groups, and the three-dimensional Heisenberg group. Our final application is to Thompson's group F.

math.GT↗