arXiv research

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.

A random group contains many quasiconvex surface subgroups.

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" grap…

A random graph of free groups contains a surface subgroup

Torelli subgroup rigidity in Out(F_N) proven for N≥4.

The study shows exponential distortion in virtually special groups containing free subgroups.

We determine the largest (i.e. smallest index) characteristic subgroup of surface groups not containing any simple loops.

The connected covering spaces of a connected and locally path-connected topological space $X$ can be classified by the conjugacy classes of those subgroups of $π_1(X,x)$ which contain an open normal subgroup of $π_1(X,x)$, when endowed with the natural quotient topology of the compact-open topology on based loops. Ther…

The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebr…

The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.

New CAT(0) groups show superexponential subgroup Dehn functions.

Let n be greater than or equal to 3. We prove that the quaternion group of order 8 is realised as a subgroup of the sphere braid group B\_n(S^2) if and only if n is even. If n is divisible by 4 then the commutator subgroup of B\_n(S^2) contains such a subgroup. Further, for all n greater than or equal to 3, B\_n(S^2) c…

The paper disproves the existence of certain subgroups with nontrivial rational abelianization.

New hyperbolic groups found with specific subgroup properties.

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.

We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.

Regular subgroups of SL3(R) are identified and ruled out.

No hyperbolic group can have an infinite chain of free subgroups of fixed rank.

Develops algorithm to find subgroups with different treatment effects in HIV patients.

New lattices in higher dimensions have dense surface subgroups.

We prove that, except for a few explicit roots of unity, the quantum image of any Johnson subgroup of the mapping class group contains an explicit free non-abelian subgroup.

We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.

Finite index subgroups of certain groups cannot act faithfully on the circle.

Exponential growth of stable subgroups in Morse geodesics.

Let A denote either the automorphism group of the free group of rank n>=4 or the mapping class group of an orientable surface of genus n>=12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of A, respectively. For a natural number N denote by G_N the Nth te…

We show that the normal closure of any periodic element of the mapping class group of a non-orientable surface whose order is greater than 2 contains the commutator subgroup, which for $g\geq 7$ is equal to the twist subgroup, and provide necessary and sufficient conditions for the normal closures of involutions to con…

We prove that $π_1(\text{Ham}(M))$ contains an infinite cyclic subgroup, where $\text{Ham}(M)$ is the Hamiltonian group of the one point blow up of ${\Bbb C}P^3$. We give a sufficient condition for the group $π_1(\text{Ham}(M))$ to contain an infinite cyclic subgroup, when $M$ is a general toric manifold.

We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for manifolds commensurable with these groups, immersed incompressible surfaces lift to embe…

We show that for any subgroup $H$ of Out($F_N$), either $H$ contains an atoroidal element or a finite index subgroup $H'$ of $H$ fixes a nontrivial conjugacy class in $F_N$. This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel-Mosher's subgroup theorem for Out($F_N$) in the setting …

The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.

We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential.

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic tur…

For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does not normally generate $F$. The question comes from a study of Hirzebruch-type inv…

In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.

The paper finds dense subgroups in certain Lie groups.

The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.

Study on kernels of SO(3) WRT representations for surfaces of genus g≥3.

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$ contains an infinite normal…

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …

We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of th…

Knots generating infinite subgroup bound rational homology balls.

The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.

We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group $SSympeo(M,ω)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,ω)$ of hamiltonian homeomorphisms introduced by Oh and Mull…

We construct CAT(0) groups containing subgroups whose Dehn functions are given by $x^s$, for a dense set of numbers $s \in [2, \infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups.

Plante-Thurston proved that every nilpotent subgroup of $\Diff^2(S^1)$ is abelian. One of our main results is a sharp converse: $\Diff^1(S^1)$ contains every finitely-generated, torsion-free nilpotent group.

We give a short proof of the following theorem of Sang-hyun Kim: if $A(Γ)$ is a right-angled Artin group with defining graph $Γ$, then $A(Γ)$ contains a hyperbolic surface subgroup if $Γ$ contains an induced subgraph $\bar{C}_n$ for some $n \geq 5$, where $\bar{C}_n$ denotes the complement graph of an $n$-cycle. Furthe…

We show that a group with a presentation satisfying the C(6) small cancellation condition cannot contain a subgroup isomorphic to F_2 X F_2.