Extensions of free groups: algebraic, geometric, and algorithmic aspects

In this work we use geometric techniques in order to study certain natural extensions of free groups, and solve several algorithmic problems on them. To this end, we consider the family of free-abelian times free groups (Zm x Fn) as a seed towards further generalization in two main directions: semidirect products, and partially commuative groups (PC-groups). The four principal projects of this thesis are the following: Direct products of free-abelian and free groups We begin by studying the structure of the groups Zm x Fn , with special emphasis on their lattice of subgroups, and their endomorphisms (for which an explicit description is given, and both injectivity and surjectiveness are characterized); to then solve on them algorithmic problems involving both subgroups (the membership problem, the finite index problem, and the subgroup and coset intersection problems), and endomorphisms (the fixed points poblem, the Whitehead problems, and the twisted-conjugacy problem). Algorithmic recognition of infinite-cyclic extensions In the first part, we prove the algorithmic undecidability of several properties (finite generability, finite presentability, abelianity, finiteness, independence, triviality) of the base group of finitely presented cyclic extensions. In particular, we see that it is not possible to decide algorithmically if a finitely presented Z-extension admits a finitely generated base group. This last result allows us to demonstrate the undecidability of the Bieri-Neumann-Strebel (BNS) invariant. In the second part, we prove the equivalence between the isomorphism problem within the subclass of unique Z-extensions, and the semi-conjugacy problem for certain type of outer automorphisms, which we characterize algorithmically. Stallings automata for free-abelian by free groups After recreating in a purely algorithmic language the classic theory of Stallings associating an automaton to each subgroup of the free group, we extend this theory to semi-direct products of the form Zm ¿ Fn. Specifically, we associate to each subgroup of Zm ¿ Fn , an automaton ("enriched" with vectors in Zm), and we see that in the finitely generated case this construction is algorithmic and allows to solve the membership problem within this family of groups. The geometric description obtained also shows (even in the case of direct products) not only that the intersection of finitely generated subgroups can be infinitely generated, but that even when it is finitely generated, the rank of the intersection can not be bound in terms of the ranks of the intersected subgroups. This fact is relevant because it denies any possible extension of the celebrated - and recently proven - Hanna-Neumann conjecture in this direction. Intersection problems for Droms groups After characterizing those partially commutative groups satisfying the Howson property, we combine the algorithmic version of the theorem of the subgroups of Kurosh given by S.V. Ivanov, with the ideas coming from our work on Zm x Fn, to prove the solvability of the subgroup and coset intersection problems within the subfamily of Droms groups (that is, those PC- groups whose subgroups are always again partially commutative).En aquest treball s'usen tècniques geomètriques per estudiar certes extensions naturals dels grups lliures, i atacar diversos problemes algorísmics sobre elles. A aquest efecte, es considera la família de grups lliure-abelians per lliure (Zm x Fn) com a punt de partida envers generalitzacions en dues direccions principals: productes semidirectes, i grups parcialment commutatius (PC-groups). Els quatre projectes principals d'aquesta tesi es descriuen a continuació. Productes directes de grups lliure-abelians per lliure. Comencem estudiant l'estructura dels grups Zm x Fn, amb especial èmfasi en el seu reticle de subgrups, i el seu monoide d'endomorfismes (per als que es dóna una descripció explícita, i es caracteritzen tant la injectivitat com l'exhaustivitat); per després resoldre sobre ells problemes algorísmics involucrant tant subgrups (el problema de la pertinença, el problema de l'índex finit, i els problemes de la intersecció de subgrups i classes laterals), com endomorfismes (el problema dels punts fixos, els problemes de Whitehead , i el problema de la "conjugació retorçada" o twisted-conjugacy problem). Reconeixement algorítmic d'extensions cícliques. A la primera part, es demostra la indecidibilitat algorísmica de diverses propietats (generabilitad finita, presentabilitad finita, abelianitat, finitud, llibertat, i trivialitat) del grup base de les extensions cícliques finitament presentades. En particular, veiem que no és possible decidir algorítmicament si una Z-extensió finitament presentada admet un grup base finitament generat. Aquest últim resultat ens permet demostrar també la indecidibilitat de l'invariant BNS (de Bieri-Neumann-Strebel). A la segona part, es demostra l'equivalència entre el problema de l'isomorfisme dins de la subclasse de Z-extensions úniques, i el problema de la semi-conjugació per a cert tipus d'automorfismes exteriors, que caracteritzem algorísmicament. Autòmats d'Stallings per a grups lliure-abelians by lliure. Després de recrear en un llenguatge purament algorísmic la teoria clàssica d'Stallings associant un autòmat a cada subgrup del grup lliure, estenem aquesta teoria a productes semidirectes de la forma Zm x Fn . Concretament associem un autòmat "enriquit" amb vectors de Zm a cada subgrup de Zm x Fn , i veiem que en el cas de subgrups finitament generats aquesta construcció és algorísmica i permet resoldre el problema de la pertinença dins d'aquesta família de grups. La descripció geomètrica obtinguda mostra a més (fins i tot en el cas de productes directes), no només que la intersecció de subgrups finitament generats pot ser infinitament generada, sinó que, fins i tot quan és finitament generada, no es pot afitar el rang de la intersecció en termes dels rangs dels subgrups intersecats. Aquest fet és rellevant perquè denega qualsevol possible extensió de la celebrada - i recentment provada - conjectura de Hanna Neumann en aquesta direcció. Problemes de la intersecció per a grups de Droms. Després de caracteritzar els grups parcialment commutatius que satisfan la propietat de Howson, combinem la versió algorísmica del teorema dels subgrups de Kurosh donada per S.V. Ivanov, amb les idees provinents del nostre treball sobre Zm x Fn, per demostrar la resolubilitat dels problemes de la intersecció de subgrups, de classes laterals (i afins) dins la subfamília de PC-grups de Droms (i.e., aquells PC-grups en que tots els subgrups son de nou parcialment commutatius)

Problemes algorísmics en grups lliure per lliure-abelià

Accèssit del Premi Évariste Galois 2012, atorgat per la Societat Catalana de MatemàtiquesEstudiem els grups lliures per lliure abelia i alguns problemes de decissió algorísmica sobre ells.. Es tracta d'estudiar els grups de la forma G=F_nxZ^m (productes directes de grups lliures per grups lliures abelians) des d'un punt de vista algorismic. La primera part del treball consistirà en fer una anàlisi de com son tots els subgrups de G, quin és el concepte adequat de "base" per aquests grups, estudiar com són tots els automorfismes i endomorfismes de G, etc. En una segona part, i amb aquesta anàlisi feta amb suficient detall, es tracta d'estudiar, entre d'altres, els següents problemes algorísmics: 1) el problema de Whitehead per a G (en les seves diverses variants, paraules, tuples de paraules, llevat conjugació o no, subgrups, tuples de subgrups, versions mixtes); 2) el problema de la conjugació i conjugació torçada per a G; 3) el problema de Brinkmann per a G; 4) càlcul del subgrup de punts fixos d'un automorfisme, i de l'estabilitzador d'un subgrup; 5) és certa la propietat de Howson per G?, càlcul de la intersecció de dos subgrups finitament generats; 6) determinació algorísmica de subgrups d'índex finit i càlcul de famílies de representants dels cosets; 7) detecció de la conjugació y subconjugació entre subgrups, malnormalitat, etc.Award-winnin

Stallings automata and applications : BGSMath graduate course

The goal of this Advanced Graduate Course is to introduce the student into the world of free groups, to show the intrinsic complexity of this algebraic structure and of the natural problems emanating from it, and to introduce the modern Stallings techniques, able to solve most of the classical problems and many new ones in a quite comprehensible and graphical way. We’ll emphasize the computational point of view, not just giving formal proofs, but also providing algorithms able to do the tasks effectively. The course will cover several classical results, like Nielsen-Schreier Theorem, Schreier index formula, Marshall Hall Theorem, membership problem, residual finiteness, the Howson property, Hanna-Neumann inequality, etc. But we plan to also introduce more advanced material in direct connection with research done in the last years: techniques for giving asymptotic estimates of properties of subgroups of the free group, enriched Stallings graphs allowing to extend results to broader families.The theory of Stallings automata provides a neat geometric representation of subgroups of the free group, and constitutes the modern —and probably the most natural and fruitful — approach to their study. Moreover, if the involved subgroups are finitely generated, then this description is finitary and very well suited for algorithmic treatment. The original result (hinted by the work of Serre, and stated in a seminal paper bytallings in 1983) interprets the subgroups of a free group FA as covering spaces of the bouquet of n circles. This mainly topological original viewpoint, has been converted into a more combinatorial one during the last decades, with the use of automata; this stresses the fact that, when restricted to finitely generated subgroups, most of the interesting problems can be resolved in a computational way, with algorithms usually more efficient and easy to understand than classical ones (more algebraic oriented). The goal of this Advanced Graduate Course is to introduce the student into the world of free groups, to show him/her the intrinsic complexity of this algebraic structure and of the natural problems emanating from it, and to introduce him/her into the modern Stallings techniques, able to solve most of the classical problems and many new ones in a quite comprehensible and graphical way (and not using much technicalities). We’ll continuously emphasise the computational point of view, not just giving formal mathematical proofs, but also providing algorithms able to do the tasks effectively by using present time computers. On one hand, this is a research topic relatively easy to get in, since the background needed is just some basic knowledge of algebra and graph theory. On the other hand, it is a hot research topic with lots of papers published since Stallings seminal paper in 1983 using these techniques, and lots that continue appearing in our days. A significant part of the publications of the three proposed lecturers use Stallings graphical techniques in an essential way. See the two recent surveys [2] and [4] from the bibliography below. The course will cover full proofs of several classical results about subgroups of free groups, like Nielsen-Schreier Theorem, Schreier index formula, Marshall Hall Theorem, membership problem, residual finiteness, the Howson property, Hanna-Neumann inequality, etc. But we plan to also introduce more advanced material in direct connection with research done in the last years: techniques for counting Stallings graphs and so for giving asymptotic estimates about properties of subgroups of the free group, enriched Stallings graphs allowing to extend results to certain groups of the form Fn X Zm or, more generally, Fn X A Zm, etc. Some connections with the theory of right-angled Artin groups will also be explained.Postprint (published version

Relative order and spectrum in free and related groups

We consider a natural generalization of the concept of order of an element in a group: an element g ¿ G is said to have order k in a subgroup H (resp., in a coset Hu) of a group G if k is the first strictly positive integer such that gk ¿ H (resp., gk ¿ Hu). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e., the set of elements of a given order) w.r.t. finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even non-recursively enumerable sets of natural numbers. Also, we take advantage of Mikhailova’s construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.The first named author was partially supported by MINECO grant PID2019-107444GA-I00 and the Basque Government grant IT974-16. The second named author acknowledges partial support from the Spanish Agencia Estatal de Investigación, through grant MTM2017-82740-P (AEI/ FEDER, UE), and also from the Graduate School of Mathematics through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). The third named author was partially supported by (Polish) Narodowe Centrum Nauki, grant UMO-2018/31/G/ST1/02681.Peer ReviewedPostprint (author's final draft

Autòmats de Stallings, un camí d'anada i tornada

En aquest article revisem algunes de les propietats fonamentals del grup lliure i fem una exposició detallada de la teoria dels autòmats de Stallings, una interpretació geomètrica dels seus subgrups que ha estat (i segueix essent) immensament fructífera, tant com a mitjà per a entendre resultats clàssics com com a font de nous resultats. N’expliquem alguns dels més rellevants.Els autors agraïm el suport parcial rebut de l’Agencia Estatal de Investigación, a través del projecte de recerca MTM2017-82740-P (AEI/FEDER, UE). El primer autor va realitzar la primera part d’aquest treball a la Universitat del País Basc (UPV/EHU) amb suport parcial del MINECO a través del projecte PID2019- 107444GA-I00, i del Govern basc amb el projecte IT974-16.Postprint (published version

The use of tricyclic antidepressants in the treatment of temporomandibular joint disorders: systematic review of the literature of the last 20 years

Many therapies have been proposed for the management of temporomandibular disorders, including the use of different drugs. However, lack of knowledge about the mechanisms behind the pain associated with this pathology, and the fact that the studies carried out so far use highly disparate patient selection criteria, mean that results on the effectiveness of the different medications are inconclusive. This study makes a systematic review of the literature published on the use of tricyclic antidepressants for the treatment of temporomandibular disorders, using the SORT criteria (Strength of recommendation taxonomy) to consider the level of scientific evidence of the different studies. Following analysis of the articles, and in function of their scientific quality, a type B recommendation is given in favor of the use of tricyclic antidepressants for the treatment of temporomandibular disorders

The use of tricyclic antidepressants in the treatment of temporomandibular joint disorders : Systematic review of the literature of the last 20 years

Many therapies have been proposed for the management of temporomandibular disorders, including the use of different drugs. However, lack of knowledge about the mechanisms behind the pain associated with this pathology, and the fact that the studies carried out so far use highly disparate patient selection criteria, mean that results on the effectiveness of the different medications are inconclusive. This study makes a systematic review of the literature published on the use of tricyclic antidepressants for the treatment of temporomandibular disorders, using the SORT criteria (Strength of recommendation taxonomy) to consider the level of scientific evidence of the different studies. Following analysis of the articles, and in function of their scientific quality, a type B recommendation is given in favor of the use of tricyclic antidepressants for the treatment of temporomandibular disorders

Fascicular Topography of the Human Median Nerve for Neuroprosthetic Surgery

One of the most sought-after applications of neuroengineering is the communication between the arm and an artificial prosthetic device for the replacement of an amputated hand or the treatment of peripheral nerve injuries. For that, an electrode is placed around or inside the median nerve to serve as interface for recording and stimulation of nerve signals coming from the fascicles that innervate the muscles responsible for hand movements. Due to the lack of a standard procedure, the electrode implantation by the surgeon is strongly based on intuition, which may result in poor performance of the neuroprosthesis because of the suboptimal location of the neural interface. To provide morphological data that can aid the neuroprosthetic surgeon with this procedure, we investigated the fascicular topography of the human median nerve along the forearm and upper arm. We first performed a description of the fascicular content and branching patterns along the length of the arm. Next we built a 3D reconstruction of the median nerve so we could analyze the fascicle morphological features in relation to the arm level. Finally, we characterized the motor content of the median nerve fascicles in the upper arm. Collectively, these results indicate that fascicular organization occurs in a short segment distal to the epicondyles and remains unaltered until the muscular branches leave the main trunk. Based on our results, overall recommendations based on electrode type and implant location can be drawn to help and aid the neuroprosthetic procedure. Invasive interfaces would be more convenient for the upper arm and the most proximal third of the forearm. Epineural electrodes seem to be most suitable for the forearm segment after fascicles have been divided from the main trunk

Electrochemical Instrumentation of an Embedded Potentiostat System (EPS) for a Programmable-System-On-a-Chip

Under the main features required on portable devices in electrochemical instrumentation is to have a small size, low power consumption, economically affordable and precision in the measurements. This paper describes the development of a programmable Embedded Potentiostat System (EPS) capable of performing electrochemical sensing over system-on-a-chip platforms. Furthermore, the study explains a circuit design and develops some validation of the entire system. The hardware validation is performed by electrochemical experiments such as Double Step Chronoamperometry (DSC), Linear Sweep Voltammetry (LSV) and Cyclic Voltammetry (CV); moreover, a comparison of the experimental signals between a commercial potentiostat and the EPS was done by analysis of errors on the response signal. Results illustrate that the EPS is capable of handling currents in the range of absolute values of 86.44 to 3000 nA and having control voltages in the range of ± 2 V. The device can support from 50 to 2000 samples per second. The EPS capabilities were compared with other compact potentiostats. The programmable EPS is an original approach which hugely reduces the hardware complexity and leads the way to create new applications for Point-of-Care or industrial developments with a reusable full electronics module

Backside polishing detector: a new protection against backside attacks

Secure chips are in permanent risk of attacks. Physical attacks usually start removing part of the package and accessing the dice by different means: laser shots, electrical or electromagnetic probes, etc. Doing this from the backside of the chip gives some advantages since no metal layers interfere between the hacker and the signals of interest. The bulk silicon is thinned from hundreds to some tens of micrometers in order to improve the performance of the attack. In this paper a backside polishing detector is presented that is sensitive to the thickness of the bulk silicon existing below the transistors, a numerical signature is generated which is related to this. The detector implements built-in self-surveillance techniques which protect it from being tampered.Postprint (published version