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We show that there exist non-unitarizable groups without non-abelian free
subgroups. Both torsion and torsion free examples are constructed. As a
by-product, we show that there exist finitely generated torsion groups with
non-vanishing first $L^2$-Betti numbers. We also relate the well-known problem
of whether every hyperbolic group is residually finite to an open question
about approximation of $L^2$-Betti numbers.

We conjecture that the word problem of Artin-Tits groups can be solved
without introducing trivial factors ss^{-1} or s^{-1}s. Here we make this
statement precise and explain how it can be seen as a weak form of
hyperbolicity. We prove the conjecture in the case of Artin-Tits groups of type
FC, and we discuss various possible approaches for further extensions, in
particular a syntactic argument that works at least in the right-angled case.

Comment: 24 pages; improved introduction from v1

Let $K$ be a non-trivial knot in the 3-sphere, $E_K$ its exterior, $G_K =
\pi_1(E_K)$ its group, and $P_K = \pi_1(\partial E_K) \subset G_K$ its
peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that
$gP_Kg^{-1} \cap P_K = \{e\}$ for any $g \in G_K$ with $g \notin P_K$, unless
$K$ is in one of the following three classes: torus knots, cable knots, and
composite knots; these are exactly the classes for which there exist annuli in
$E_K$ attached to $T_K$ which are not boundary parallel (Theorem 1 and
Corollary 2). More generally, we characterise malnormal peripheral subgroups in
the fundamental group of a compact orientable irreducible 3-manifold with
boundary a non-empty union of tori (Theorem 3). Proofs are written with
non-expert readers in mind. Half of our paper (Sections 7 to 10) is a reminder
of some three-ma...

Comment: 26 pages, 31 figures

A commutative loop is Jordan if it satisfies the identity $x^2 (y x) = (x^2
y) x$. Using an amalgam construction and its generalizations, we prove that a
nonassociative Jordan loop of order $n$ exists if and only if $n\geq 6$ and
$n\neq 9$. We also consider whether powers of elements in Jordan loops are
well-defined, and we construct an infinite family of finite simple
nonassociative Jordan loops.

Comment: This paper has been withdrawn

We prove that pure braid groups of closed surface are almost-direct products
of residually torsion free nilpotent groups and hence residually torsion free
nilpotent. As a Corollary, we prove also that braid groups on 2 strands of
closed surfaces are residually nilpotent.

We prove that if $\pi$ is a recursive set of primes, then pointlike sets are
decidable for the pseudovariety of semigroups whose subgroups are $\pi$-groups.
In particular, when $\pi$ is the empty set, we obtain Henckell's decidability
of aperiodic pointlikes. Our proof, restricted to the case of aperiodic
semigroups, is simpler than the original proof.