arXiv subjects
Publications and source records attributed to Jesús Hernández Hernández.
We give a criterion for property $P_{\text{naive}}$ for subgroups of orientable big mapping class groups and apply it to mapping class groups of non-orientable infinite type surfaces through the orientation double cover.
Let $N$ be a connected closed non-orientable surface of genus at least 6. In this work we prove that there exists a finite subgraph $\mathfrak{X}$ (Irmak's finite rigid set from ``Elmas Irmak. Exhausting curve complexes by finite rigid sets on nonorientable surfaces. J.Topol.Anal., 16(2):261--289, 2024'') such that any graph endomorphism $φ$ of $\mathcal{C}(N)$ whose restriction to $\mathfrak{X}$ is (locally) injective, $φ$ is induced by a homeomorphism of $N$. To prove this, we first prove that $\mathfrak{X}$ and its rigid expansions exhaust $\mathcal{C}(N)$.
In this paper we obtain explicit linear upper bounds for the virtually cyclic dimension of normally poly-surface and normally poly-free groups. Our approach is based on a structural study of the balanced property (Lück's Condition~C), which provides structural control over commensurators of virtually cyclic subgroups. We prove general stability results showing that the balanced property is preserved under suitable short exact sequences, direct limits, and acylindrical graph of groups decompositions. As applications, we establish that normally poly-hyperbolic groups, normally poly-free groups, and normally poly-surface groups are balanced. These classes include, in particular, pure braid groups of surfaces with non-empty boundary, Artin groups of FC-type, right-angled Artin groups, and fundamental groups of mapping tori of surface homeomorphisms.
We show that the open genus 2 handlebody admits uncountably-many fibrations over the circle with fiber homeomorphic to the Cantor tree surface with non-conjugate monodromies in the mapping class group. The construction generalizes to produce uncountably many fibrations of other tame 3-manifolds with infinite type fibers, including the blooming Cantor tree and many other types of surfaces.
This work is the extension of the results by the author in [7] and [6] for low-genus surfaces. Let $S$ be an orientable, connected surface of finite topological type, with genus $g \leq 2$, empty boundary, and complexity at least $2$; as a complement of the results of [6], we prove that any graph endomorphism of the curve graph of $S$ is actually an automorphism. Also, as a complement of the results in [6] we prove that under mild conditions on the complexity of the underlying surfaces any graph morphism between curve graphs is induced by a homeomorphism of the surfaces. To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid sets in [2]. Similarly to [7], a consequence of our proof is that Aramayona and Leininger's rigid set also exhausts the curve graph via rigid expansions, and the combinatorial rigidity results follow as an immediate consequence, based on the results in [6].
We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface $Σ$ on itself. Our main results are: (1) All conjugacy classes of $MCG(Σ)$ are meager for every $Σ$, (2) $MCG(Σ)$ has a somewhere dense conjugacy class if and only if $Σ$ has at most two maximal ends and no non-displaceable finite-type subsurfaces, (3) $MCG(Σ)$ has a dense conjugacy class if and only if $Σ$ has a unique maximal end and no non-displaceable finite-type subsurfaces. Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Truss.
In this work we compute the first integral cohomology of the pure mapping class group of a non-orientable surface of infinite topological type and genus at least 3. To this purpose, we also prove several other results already known for orientable surfaces such as the existence of an Alexander method, the fact that the mapping class group is isomorphic to the automorphism group of the curve graph along with the topological rigidity of the curve graph, and the structure of the pure mapping class group as both a Polish group and a semi-direct product.
Let $S_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We identify a finite rigid subgraph $X_{g,n}$ of the pants graph $\mathcal P (S_{g,n})$, that is, a subgraph with the property that any simplicial embedding of $X_{g,n}$ into any pants graph $\mathcal P (S_{g',n'})$ is induced by an embedding $S_{g,n}\to S_{g',n'}$. This extends results of the third author for the case of genus zero surfaces.
Let $ϕ:\mathcal{C}(S)\to\mathcal{C}(S')$ be a simplicial isomorphism between the curve graphs of two infinite-type surfaces. In this paper we show that in this situation $S$ and $S'$ are homeomorphic and $ϕ$ is induced by a homeomorphism $h:S\to S'$.
Let $S_{1}$ and $S_{2}$ be connected orientable surfaces of genus $g_{1}, g_{2} \geq 3$, $n_{1},n_{2} \geq 0$ punctures, and empty boundary. Let also $φ: \mathcal{HT}(S_{1}) \rightarrow \mathcal{HT}(S_{2})$ be an edge-preserving alternating map between their Hatcher-Thurston graphs. We prove that $g_{1} \leq g_{2}$ and that there is also a multicurve of cardinality $g_{2} - g_{1}$ contained in every element of the image. We also prove that if $n_{1} = 0$ and $g_{1} = g_{2}$, then the map $\widetildeφ$ obtained by filling the punctures of $S_{2}$, is induced by a homeomorphism of $S_{1}$.
Suppose $S_{1}$ and $S_{2}$ are orientable surfaces of finite topological type such that $S_{1}$ has genus at least $3$ and the complexity of $S_{1}$ is an upper bound of the complexity of $S_{2}$. Let $φ: \mathcal{C}(S_{1}) \rightarrow \mathcal{C}(S_{2})$ be an edge-preserving map; then $S_{1}$ is homeomorphic to $S_{2}$, and in fact $φ$ is induced by a homeomorphism. To prove this, we use several simplicial properties of rigid expansions, which we prove here.
For an orientable surface $S$ of finite topological type with genus $g \geq 3$, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of $S$. The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.
Let S be any orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the topological ends of S carry genus, we show that all elements in the automorphism groups Aut(C(S)), Aut(N(S)) and Aut(G(S)) are geometric, i.e. these groups are naturally isomorphic to the extended mapping class group MCG*(S) of the infinite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S))