Related papers: C(6) groups do not contain F_2 X F_2

We give a new proof of the main theorem in the theory of C(6) small cancellation complexes. We prove the fundamental theorem of cubical small cancellation theory for C(9) cubical small cancellation complexes.

We prove that non-trivial representations of the alternating group $A_n$ are reducible over a primitive proper subgroup which is isomorphic to some alternating group $A_m$.

We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear,…

We prove the vanishing of certain low degree cohomologies of some induced representations. As an application, we determine certain low degree cohomologies of congruence groups.

We address the following question. For which finitely generated pro-$p$ groups the comparison map $\phi^2:H_{cont}^{2}(P,\F_p) \to H_{disc}{2}(P,\F_p)$ is an isomorphism? We prove that if $P$ is not finitely presented then $\phi^2$ is not…

A conjecture of Roseberger asserts that every generalised triangle group either is virtually soluble or contains a non-abelian free subgroup. Modulo two exceptional cases, we verify this conjecture for generalised triangle groups of type…

We introduce the representation category $\mathscr{C}({\bf G})$ for a connected reductive algebraic group ${\bf G}$ which is defined over a finite field $\mathbb{F}_q$ of $q$ elements. We show that this category has many good properties for…

For a group $G$, $\mathcal{F}_G$ denotes the set of all non-empty finite subsets of $G$. We extend the finitary coarse structure of $G$ from $G\times G$ to $\mathcal{F}_G\times \mathcal{F}_G$ and say that a macro-uniform mapping $f:…

For a non-cyclic free group $F$, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.

We prove that finitely generated (not necessarily finitely presented) $C'(\frac{1}{33})$-groups are bi-exact. This is a new class of bi-exact groups.

In this paper we prove that there does not exist a subgroup $H$ of a finite group $G$ such that the number of isomorphism classes of normalized right transversals of $H$ in $G$ is four.

We show that a finitely presented group virtually admits a planar Cayley graph if and only if it is asymptotically minor-excluded, partially answering a conjecture of Georgakopoulos and Papasoglu in the affirmative.

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact…

If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = {\pi}_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F'…

We construct an uncountable family of groups of type $FP$. In contrast to every previous construction of non-finitely presented groups of type $FP$ we do not use Morse theory on cubical complexes; instead we use Gromov's graphical small…

Let G be a Frobenius group with the Frobenius kernel K. Suppose that G contains a nontrival subgroup D \subseteq K such that the normalizer N_G(D) \not\subseteq K. When D is no 2-group, Flavell proved, without using character theory, that K…

Given a global exponent $E$ for a black box group $\mathsf{Y}$ encrypting ${\rm SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an unknown finite field of unknown odd characteristic, we construct, in probabilistic time polynomial in $\log E$, the…

A cover of a finite group $G$ is a family of proper subgroups of $G$ whose union is $G$, and a cover is called minimal if it is a cover of minimal cardinality. A partition of $G$ is a cover such that the intersection of any two of its…

Let $N_1$ (resp., $N_2$) be the normal closure of a finite symmetrized set $R_1$ (resp., $R_2$) of a finitely generated free group $F = F(A)$. It is well-known that if $R_i$ satisfies the condition C(6), then the conjugacy problem is…