Injective Homomorphism Research Articles

Rigidity of the Torelli subgroup in $\mathrm{Out}(F_{N})$

Let N\ge 4 . We prove that every injective homomorphism from the Torelli subgroup \textup{IA}_{N} to \textup{Out}(F_{N}) differs from the inclusion by a conjugation in \textup{Out}(F_{N}) . This applies more generally to the following subgroups of \textup{Out}(F_{N}) : every finite-index subgroup of \textup{Out}(F_{N}) (recovering a theorem of Farb and Handel); every subgroup of \textup{Out}(F_{N}) that contains a finite-index subgroup of one of the groups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) ; every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup of \textup{Out}(F_{N}) –those are subgroups that contain sufficiently many twists in an appropriate sense.Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in \textup{Out}(F_{N}) ; it also implies that all subgroups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) are co-Hopfian.We also prove the same rigidity statement for subgroups of \textup{Out}(F_{3}) which contain a power of every Nielsen transformation. This shows, in particular, that \textup{Out}(F_{3}) and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the N=3 case.

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let M = ( M , ω ) M=(M,\omega ) be either S 2 × S 2 S^2 \times S^2 or C P 2 # C P 2 ¯ \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form ω \omega . Suppose a finite cyclic group Z n \mathbb {Z}_n is acting effectively on ( M , ω ) (M,\omega ) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z n ↪ H a m ( M , ω ) \mathbb {Z}_n\hookrightarrow Ham(M,\omega ) . In this paper, we investigate the homotopy type of the group S y m p Z n ( M , ω ) Symp^{\mathbb {Z}_n}(M,\omega ) of equivariant symplectomorphisms. We prove that for some infinite families of Z n \mathbb {Z}_n actions satisfying certain inequalities involving the order n n and the symplectic cohomology class [ ω ] [\omega ] , the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J J -holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4 4 -manifolds, and on the Chen-Wilczyński classification of smooth Z n \mathbb {Z}_n -actions on Hirzebruch surfaces.

Big Mapping Class Groups and the Co-Hopfian Property

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

On categories of faithful quandles with surjective or injective quandle homomorphisms

On categories of faithful quandles with surjective or injective quandle homomorphisms

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Let N be a connected nonorientable surface with or without boundary and punctures, and j\colon S\rightarrow N be the orientation double covering. It has previously been proved that j induces an embedding \iota\colon\mathrm{Mod}(N)\hookrightarrow\mathrm{Mod}(S) with one exception. In this paper, we prove that the injective homomorphism \iota is a quasi-isometric embedding. The proof is based on the semihyperbolicity of \mathrm{Mod}(S) , which has already been established. We also prove that the embedding \mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F) induced by an inclusion of a pair of possibly nonorientable surfaces F' \subset F is a quasi-isometric embedding.

Realizability and tameness of fusion systems

AbstractA saturated fusion system over a finite ‐group is a category whose objects are the subgroups of and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over is realized by a finite group if is a Sylow ‐subgroup of and morphisms in the category are those induced by conjugation in . One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the classification of finite simple groups, but we also give more precise formulations whose proof is independent of the classification.

Homomorphisms and inverse homomorphisms on graph-walking automata

Graph-walking automata analyze an input graph by moving between its nodes, following the edges. This paper investigates the effect of node-replacement graph homomorphisms and inverse homomorphisms on recognizability by these automata. For deterministic graph-walking automata, it is shown that the family of graph languages they recognize is closed under inverse homomorphisms: for an n-state automaton, the inverse homomorphic images of the graphs it accepts can be recognized by an automaton with at most kn+1 states, where k is the number of labels of edge end-points in the pre-image graphs. At the same time, it is proved that in the worst case these inverse homomorphic images require a deterministic graph-walking automaton with at least kn states. The upper bound kn+1 also holds for nondeterministic graph-walking automata. The second result is that already for tree-walking automata, both deterministic and nondeterministic, the families they recognize are not closed under injective homomorphisms. Here the proof is based on a new homomorphic characterization of regular tree languages: every regular tree language is representable as h−1(g(all trees)), for some injective node-replacement homomorphisms g and h.

Hall algebras and quantum symmetric pairs of Kac-Moody type

We extend our ıHall algebra construction from acyclic to arbitrary ıquivers, where the ıquiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal ıquantum group of Kac-Moody type arising from quantum symmetric pairs to the ıHall algebra associated to a virtually acyclic ıquiver.

Канторовость квазиунитарных полигонов над вполне (0-)простыми полугруппами

An universal algebra $A$ is called cantorian if for any algebra $B$ of the same signature, the existence of injective homomorphisms $A\to B$ and $B \to A$ implies an isomorphism of algebras $A$ and $B$. A right act $X$ over a semigroup $S$ is called quasiunitary if $X=XS$. We prove that every quasiunitary act over a completely simple semigroup and also every quasiunitary act with zero over a completely 0-simple semigroup are cantorian.

Complexity of Injective Homomorphisms to Small Tournaments, and of Injective Oriented Colourings

Several possible definitions of local injectivity for a homomorphism of an oriented graph G to an oriented graph H are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when G is given and H is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.

Categorically Closed Unipotent Semigroups

Let C be a class of T1 topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup X is C-closed if X is closed in any topological semigroup Y∈C that contains X as a discrete subsemigroup; X is injectively C-closed if for any injective homomorphism h:X→Y to a topological semigroup Y∈C the image h[X] is closed in Y. A semigroup X is unipotent if it contains a unique idempotent. It is proven that a unipotent commutative semigroup X is (injectively) C-closed if and only if X is bounded and nonsingular (and group-finite). This characterization implies that for every injectively C-closed unipotent semigroup X, the center Z(X) is injectively C-closed.

Dominance phenomena: mutation, scattering and cluster algebras

An exchange matrix B B dominates an exchange matrix B ′ B’ if the signs of corresponding entries weakly agree, with the entry of B B always having weakly greater absolute value. When B B dominates B ′ B’ , interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for B B often refines the mutation fan for B ′ B’ ; the scattering (diagram) fan for B B often refines the scattering fan for B ′ B’ ; and there is often an injective homomorphism from the principal-coefficients cluster algebra for B ′ B’ to the principal-coefficients cluster algebra for B B , preserving g \mathbf {g} -vectors and sending the set of cluster variables for B ′ B’ (or an analogous larger set) into the set of cluster variables for B B (or an analogous larger set). The scope of the description “often” is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena.

On the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields

In the present paper, we study the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of anabelian geometry. Let us recall that it is well-known that the natural homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the absolute Galois group of the given mixed-characteristic local field is injective. One main result of the present paper is that if the mixed-characteristic local field satisfies certain conditions, then the set of conjugates of the image of this injective homomorphism in the outer automorphism group is infinite, which thus implies that the image of this injective homomorphism is not normal in the outer automorphism group. In particular, one may conclude that it is impossible to establish a functorial group-theoretic reconstruction, from the absolute Galois group, of the “field-theoretic” subgroup, i.e., the image of this injective homomorphism, of the outer automorphism group.

On the Cantor and Hilbert cube frames and the Alexandroff-Hausdorff theorem

The aim of this work is to give a point-free description of the Cantor set . It can be shown (see, e.g. [18] ) that the Cantor set is homeomorphic to the p -adic integers Z p : = { x ∈ Q p : | x | p ≤ 1 } for every prime number p . To give a point-free description of the Cantor set, we specify the frame of Z p by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of Z p and thus we think of them as the basic generators; on this poset we impose some relations and then the resulting quotient is the frame of the Cantor set L ( Z p ) . A topological characterization of it is given by Brouwer's Theorem [7] : The Cantor set is the unique totally disconnected, compact metric space with no isolated points. We prove that L ( Z p ) is a spatial frame whose space of points is homeomorphic to Z p . In particular, we show with point-free arguments that L ( Z p ) is 0-dimensional, (completely) regular, compact, and metrizable (it admits a countably generated uniformity). Moreover, we show that the frame L ( Z p ) satisfies c b d L ( Z p ) ( 0 ) = 0 , where c b d L ( Z p ) : L ( Z p ) → L ( Z p ) , defined by c b d L ( Z p ) ( a ) = ⋀ { x ∈ L ( Z p ) | x ≤ a and ( x → a ) = a } is the Cantor-Bendixson derivative (see, e.g. [19] ). It follows that a frame L is isomorphic to L ( Z p ) if and only if L is a 0-dimensional compact regular metrizable frame with c b d L ( 0 ) = 0 . Finally, we give a point-free counterpart of the Hausdorff-Alexandroff Theorem which states that every compact metric space is a continuous image of the Cantor space (see, e.g. [1] and [12] ). We prove the point-free analogue: if L is a compact metrizable frame, then there is an injective frame homomorphism from L into L ( Z 2 ) .

On the second realization for the positive part of $$U_q(\widehat{sl_2})$$ of equitable type

The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type $U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ are introduced and studied. A presentation for $U_q^{T,+}$ is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-operators are considered, from which an explicit injective homomorphism from $U_q^{T,+}$ to a subalgebra of Drinfeld's second realization (current algebra) of $U_q(\widehat{sl_2})$ is derived, and the comodule algebra structure of $U_q^{T,+}$ is characterized. The central extension of $U_q^{T,+}$ and its relation with Drinfeld's second realization of $U_q(\widehat{gl_2})$ is also described using the framework of Freidel-Maillet algebras.

Classical and variational Poisson cohomology

We prove that, for a Poisson vertex algebra \({\cal V}\), the canonical injective homomorphism of the variational cohomology of \({\cal V}\) to its classical cohomology is an isomorphism, provided that \({\cal V}\), viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.

Operadic lifts of the algebra of indexing systems

Abstract For a given finite group G , the homotopy category of N ∞ G -operads is equivalent to a finite lattice , and G varies, there are various image constructions between these lattices. In this paper, we explain how to lift this algebraic structure back to the operad level. We show that lattice joins and meets correspond to N ∞ coproducts and products, and we show that the image constructions correspond to N ∞ induction, restriction, and coinduction constructions, at least when taken along an injective homomorphism. We also prove that a N ∞ variant of the Boardman-Vogt tensor product lifts the join. Our result does not resolve Blumberg and Hill's conjecture that the ordinary tensor product of suitably cofibrant N ∞ operads models the join, but it does imply a closely related result. If O and P are operads, then an algebra over the Boardman-Vogt tensor product O ⊗ P is equipped with a pair of interchanging O and P -actions. We prove that under mild hypotheses on a N ∞ operad O , every orthogonal O -ring spectrum is weakly equivalent to a spectrum over an operad O ′ ≃ O that interchanges with itself.

Parameterized Counting of Partially Injective Homomorphisms

We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and #mathsf {W[1]}-complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.

Finite rigid sets in flip graphs

We show that for most pairs of surfaces, there exists a finite subgraph of the flip graph of the first surface so that any injective homomorphism of this finite subgraph into the flip graph of the second surface can be extended uniquely to an injective homomorphism between the two flip graphs. Combined with a result of Aramayona-Koberda-Parlier, this implies that any such injective homomorphism of this finite set is induced by an embedding of the surfaces. We also include images of several flip graphs in an appendix.

Hall Algebras and Quantum Symmetric Pairs II: Reflection Functors

We extend our $\imath$Hall algebra construction from acyclic to arbitrary $\imath$quivers, where the $\imath$quiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal $\imath$quantum group of Kac-Moody type arising from quantum symmetric pairs to the $\imath$Hall algebra associated to a virtually acyclic $\imath$quiver.