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Patrick Dehornoy

Publications and source records attributed to Patrick Dehornoy.

46 records · Page 3Linked to original sources
arXiv2005

Still another approach to the braid ordering

We develop a new approach to the linear ordering of the braid group $B\_n$, based on investigating its restriction to the set $\Div(Δ\_n^d)$ of all divisors of $Δ\_n^d$ in the monoid $B\_\infty^+$, i.e., to positive $n$-braids whose normal form has length at most $d$. In the general case, we compute several numerical parameters attached with the finite orders $(\Div(Δ\_n^d), <)$. In the case of 3 strands, we moreover give a complete description of the increasing enumeration of $(\Div(Δ\_3^d), <)$. We deduce a new and specially direct construction of the ordering on $B\_3$, and a new proof of the result that its restriction to $B\_3^+$ is a well-ordering of ordinal type $ω^ω$.

math.GR↗
arXiv2005

Geometric presentations for Thompson's groups

We prove that Thompson's groups $F$ and $V$ are the geometry groups of associativity, and of associativity together with commutativity, respectively. We deduce new presentations of $F$ and $V$. These presentations lead to considering a certain subgroup of $V$ and an extension of this subgroup. We prove that the latter are the geometry groups of associativity together with the law $x(yz) = y(xz)$, and of associativity together with a twisted version of this law involving self-distributivity, respectively.

math.GR↗
arXiv2004

On word reversing in braid groups

It has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear.

math.GR↗
arXiv2003

Braids in trivial braid diagrams

We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph and a continuity argument.

math.GT↗
arXiv2001

Complete positive group presentations

A combinatorial property of prositive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show how to directly read several properties of the associated monoid and group from a complete presentation: cancellativity or existence of common multiples in the case of the monoid, or isoperimetric inequality in the case of the group. In particular, we obtain a new criterion for recognizing that a monoid embeds in a group of fractions. Typical presentations eligible for the current approach are the standard presentations of the Artin groups and the Heisenberg group.

math.GR↗
arXiv2001

Thin groups of fractions

A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are satisfied \cite{Dgk}. Here we investigate a wider class of groups of fractions, called {\it thin}, which are those associated with monoids where minimal common multiples exist, but they are not necessarily unique. Also, we allow units in the involved monoids. The main results are that all thin groups of fractions satisfy a quadratic isoperimetric inequality, and that, under some additional hypotheses, they admit an automatic structure.

math.GR↗
arXiv2001

Homology of Gaussian groups

We describe new combinatorial methods for constructing an explicit free resolution of Z by ZG-modules when G is a group of fractions of a monoid where enough least common multiples exist (``locally Gaussian monoid''), and, therefore, for computing the homology of G. Our constructions apply in particular to all Artin groups of finite Coxeter type, so, as a corollary, they give new ways of computing the homology of these groups.

math.GR↗
arXiv2001

Groupes de Garside

Define a Garside monoid to be a cancellative monoid where right and left lcm's exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garside monoid. The family of Garside groups contains the braid groups, all spherical Artin groups, and various generalizations previously considered (In particular, the Garside groups considered in [1,7] are special cases of those considered here; the latter had been called "small Gaussian" in earlier works; the name has been changed in order to uniformize the terminology with works by Bessis, Charney, Michel, and other authors). Here we prove that Garside groups are biautomatic, and that being a Garside group is a recursively enumerable property, i.e., there exists an algorithm constructing the (infinite) list of all small Gaussian groups. The latter result relies on an effective, tractable method for recognizing those presentations that define a Garside monoid.

math.GR↗
arXiv2001

Study of an identity

We solve the word problem of the identity $x(yz) = (xy)(yz)$ by investigating a certain group describing the geometry of that identity. We also construct a concrete realization of the free system of rank~1 relative to the above identity

math.LO↗