❤️ Dehn function 🦒

"In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of the length of that relation (see pp. 79-80 in ). The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive (see Theorem 2.1 in ). The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface. History The idea of an isoperimetric function for a finitely presented group goes back to the work of Max Dehn in 1910s. Dehn proved that the word problem for the standard presentation of the fundamental group of a closed oriented surface of genus at least two is solvable by what is now called Dehn's algorithm. A direct consequence of this fact is that for this presentation the Dehn function satisfies Dehn(n) ≤ n. This result was extended in 1960s by Martin Greendlinger to finitely presented groups satisfying the C'(1/6) small cancellation condition.Martin Greendlinger, Dehn's algorithm for the word problem. Communications on Pure and Applied Mathematics, vol. 13 (1960), pp. 67-83. The formal notion of an isoperimetric function and a Dehn function as it is used today appeared in late 1980s - early 1990s together with the introduction and development of the theory of word-hyperbolic groups. In his 1987 monograph "Hyperbolic groups"M. Gromov, Hyperbolic Groups in: "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263\. . Gromov proved that a finitely presented group is word-hyperbolic if and only if it satisfies a linear isoperimetric inequality, that is, if and only if the Dehn function of this group is equivalent to the function f(n) = n. Gromov's proof was in large part informed by analogy with filling area functions for compact Riemannian manifolds where the area of a minimal surface bounding a null-homotopic closed curve is bounded in terms of the length of that curve. The study of isoperimetric and Dehn functions quickly developed into a separate major theme in geometric group theory, especially since the growth types of these functions are natural quasi-isometry invariants of finitely presented groups. One of the major results in the subject was obtained by Sapir, Birget and Rips who showedM. Sapir, J.-C. Birget, E. Rips. Isoperimetric and isodiametric functions of groups. Annals of Mathematics (2), vol 156 (2002), no. 2, pp. 345-466. that most "reasonable" time complexity functions of Turing machines can be realized, up to natural equivalence, as Dehn functions of finitely presented groups. Formal definition Let : G=\langle XR\rangle\qquad (*) be a finite group presentation where the X is a finite alphabet and where R ⊆ F(X) is a finite set of cyclically reduced words. =Area of a relation= Let w ∈ F(X) be a relation in G, that is, a freely reduced word such that w = 1 in G. Note that this is equivalent to saying that w belongs to the normal closure of R in F(X), that is, there exists a representation of w as :w=u_1r_1u_1^{-1}\cdots u_m r_mu_{m}^{-1} \text{ in } F(X), (♠) where m ≥ 0 and where ri ∈ R± 1 for i = 1, ..., m. For w ∈ F(X) satisfying w = 1 in G, the area of w with respect to (∗), denoted Area(w), is the smallest m ≥ 0 such that there exists a representation (♠) for w as the product in F(X) of m conjugates of elements of R± 1. A freely reduced word w ∈ F(X) satisfies w = 1 in G if and only if the loop labeled by w in the presentation complex for G corresponding to (∗) is null-homotopic. This fact can be used to show that Area(w) is the smallest number of 2-cells in a van Kampen diagram over (∗) with boundary cycle labelled by w. =Isoperimetric function= An isoperimetric function for a finite presentation (∗) is a monotone non-decreasing function :f: \mathbb N\to [0,\infty) such that whenever w ∈ F(X) is a freely reduced word satisfying w = 1 in G, then :Area(w) ≤ f(w), where w is the length of the word w. =Dehn function= Then the Dehn function of a finite presentation (∗) is defined as :{\rm Dehn}(n)=\max{{\rm Area}(w): w=1 \text{ in } G, w\le n, w \text{ freely reduced}.} Equivalently, Dehn(n) is the smallest isoperimetric function for (∗), that is, Dehn(n) is an isoperimetric function for (∗) and for any other isoperimetric function f(n) we have :Dehn(n) ≤ f(n) for every n ≥ 0\. =Growth types of functions= Because Dehn functions are usually difficult to compute precisely, one usually studies their asymptotic growth types as n tends to infinity. For two monotone-nondecreasing functions :f,g: \mathbb N\to [0,\infty) one says that f is dominated by g if there exists C ≥1 such that : f(n)\le Cg(Cn+C)+Cn+C for every integer n ≥ 0\. Say that f ≈ g if f is dominated by g and g is dominated by f. Then ≈ is an equivalence relation and Dehn functions and isoperimetric functions are usually studied up to this equivalence relation. Thus for any a,b > 1 we have an ≈ bn. Similarly, if f(n) is a polynomial of degree d (where d ≥ 1 is a real number) with non-negative coefficients, then f(n) ≈ nd. Also, 1 ≈ n. If a finite group presentation admits an isoperimetric function f(n) that is equivalent to a linear (respectively, quadratic, cubic, polynomial, exponential, etc.) function in n, the presentation is said to satisfy a linear (respectively, quadratic, cubic, polynomial, exponential, etc.) isoperimetric inequality. Basic properties *If G and H are quasi-isometric finitely presented groups and some finite presentation of G has an isoperimetric function f(n) then for any finite presentation of H there is an isoperimetric function equivalent to f(n). In particular, this fact holds for G = H, where the same group is given by two different finite presentations. *Consequently, for a finitely presented group the growth type of its Dehn function, in the sense of the above definition, does not depend on the choice of a finite presentation for that group. More generally, if two finitely presented groups are quasi-isometric then their Dehn functions are equivalent. *For a finitely presented group G given by a finite presentation (∗) the following conditions are equivalent: **G has a recursive Dehn function with respect to (∗). **There exists a recursive isoperimetric function f(n) for (∗). **The group G has solvable word problem. ::In particular, this implies that solvability of the word problem is a quasi- isometry invariant for finitely presented groups. *Knowing the area Area(w) of a relation w allows to bound, in terms of w, not only the number of conjugates of the defining relations in (♠) but the lengths of the conjugating elements ui as well. As a consequence, it is knownS. M. Gersten, Isoperimetric and isodiametric functions of finite presentations. Geometric group theory, Vol. 1 (Sussex, 1991), pp. 79-96, London Math. Soc. Lecture Note Ser., 181, Cambridge University Press, Cambridge, 1993.Juan M. Alonso, Inégalités isopérimétriques et quasi-isométries. Comptes Rendus de l'Académie des Sciences, Série I, vol. 311 (1990), no. 12, pp. 761-764. that if a finitely presented group G given by a finite presentation (∗) has computable Dehn function Dehn(n), then the word problem for G is solvable with non- deterministic time complexity Dehn(n) and deterministic time complexity Exp(Dehn(n)). However, in general there is no reasonable bound on the Dehn function of a finitely presented group in terms of the deterministic time complexity of the word problem and the gap between the two functions can be quite large. Examples *For any finite presentation of a finite group G we have Dehn(n) ≈ n.Martin R. Bridson. The geometry of the word problem. Invitations to geometry and topology, pp. 29-91, Oxford Graduate Texts in Mathematics, 7, Oxford University Press, Oxford, 2002. . *For the closed oriented surface of genus 2, the standard presentation of its fundamental group :G=\langle a_1,a_2,b_1,b_n[a_1,b_1][a_2,b_2]=1\rangle :satisfies Dehn(n) ≤ n and Dehn(n) ≈ n. *For every integer k ≥ 2 the free abelian group \mathbb Z^k has Dehn(n) ≈ n2. *The Baumslag-Solitar group :B(1,2)=\langle a,b b^{-1}ab=a^2\rangle :has Dehn(n) ≈ 2n (see S. M. Gersten, Dehn functions and l1-norms of finite presentations. Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), pp. 195-224, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992. .). *The 3-dimensional discrete Heisenberg group :H_3=\langle a,b, t [a,t]=[b,t]=1, [a,b]=t^2 \rangle :satisfies a cubic but no quadratic isoperimetric inequality. *Higher- dimensional Heisenberg groups :H_{2k+1}=\langle a_1,b_1,\dots, a_k,b_k,t [a_i,b_i]=t, [a_i,t]=[b_i,t]=1, i=1,\dots, k, [a_i,b_j]=1, i e j\rangle, :where k ≥ 2, satisfy quadratic isoperimetric inequalities.D. Allcock, An isoperimetric inequality for the Heisenberg groups. Geometric and Functional Analysis, vol. 8 (1998), no. 2, pp. 219-233. *If G is a "Novikov-Boone group", that is, a finitely presented group with unsolvable word problem, then the Dehn function of G growths faster than any recursive function. *For the Thompson group F the Dehn function is quadratic, that is, equivalent to n2 (see V. S. Guba, The Dehn function of Richard Thompson's group F is quadratic. Inventiones Mathematicae, vol. 163 (2006), no. 2, pp. 313-342.). *The so- called Baumslag-Gersten group ::G=\langle a, t (t^{-1}a^{-1} t) a (t^{-1} at)=a^2\rangle :has a Dehn function growing faster than any fixed iterated tower of exponentials. Specifically, for this group ::Dehn(n) ≈ exp(exp(exp(...(exp(1))...))) :where the number of exponentials is equal to the integral part of log2(n) (see A. N. Platonov, An isoparametric function of the Baumslag-Gersten group. (in Russian.) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2004, , no. 3, pp. 12-17; translation in: Moscow University Mathematics Bulletin, vol. 59 (2004), no. 3, pp. 12-17 (2005).). Known results *A finitely presented group is word-hyperbolic group if and only if its Dehn function is equivalent to n, that is, if and only if every finite presentation of this group satisfies a linear isoperimetric inequality. *Isoperimetric gap: If a finitely presented group satisfies a subquadratic isoperimetric inequality then it is word-hyperbolic.A. Yu. Olʹshanskii. Hyperbolicity of groups with subquadratic isoperimetric inequality. International Journal of Algebra and Computation, vol. 1 (1991), no. 3, pp. 281-289\. B. H. Bowditch. A short proof that a subquadratic isoperimetric inequality implies a linear one. Michigan Mathematical Journal, vol. 42 (1995), no. 1, pp. 103-107\. Thus there are no finitely presented groups with Dehn functions equivalent to nd with d ∈ (1,2). *Automatic groups and, more generally, combable groups satisfy quadratic isoperimetric inequalities.D. B. A. Epstein, J. W. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston. Word Processing in Groups. Jones and Bartlett Publishers, Boston, MA, 1992. *A finitely generated nilpotent group has a Dehn function equivalent to nd where d ≥ 1 and all positive integers d are realized in this way. Moreover, every finitely generated nilpotent group G admits a polynomial isoperimetric inequality of degree c + 1, where c is the nilpotency class of G.S. M. Gersten, D. F. Holt, T. R. Riley, Isoperimetric inequalities for nilpotent groups. Geometric and Functional Analysis, vol. 13 (2003), no. 4, pp. 795-814\. *The set of real numbers d ≥ 1, such that there exists a finitely presented group with Dehn function equivalent to nd, is dense in the interval [2,\infty).N. Brady and M. R. Bridson, There is only one gap in the isoperimetric spectrum. Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1053-1070. *If all asymptotic cones of a finitely presented group are simply connected, then the group satisfies a polynomial isoperimetric inequality.M. Gromov, Asymptotic invariants of infinite groups, in: "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1-295. *If a finitely presented group satisfies a quadratic isoperimetric inequality, then all asymptotic cones of this group are simply connected.P. Papasoglu. On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality. Journal of Differential Geometry, vol. 44 (1996), no. 4, pp. 789-806. *If (M,g) is a closed Riemannian manifold and G = π1(M) then the Dehn function of G is equivalent to the filling area function of the manifold.J. Burillo and J. Taback. Equivalence of geometric and combinatorial Dehn functions. New York Journal of Mathematics, vol. 8 (2002), pp. 169-179. *If G is a group acting properly discontinuously and cocompactly by isometries on a CAT(0) space, then G satisfies a quadratic isoperimetric inequality.M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. ; Remark 1.7, p. 444. In particular, this applies to the case where G is the fundamental group of a closed Riemannian manifold of non-positive sectional curvature (not necessarily constant). *The Dehn function of SL(m, Z) is at most exponential for any m ≥ 3.E. Leuzinger. On polyhedral retracts and compactifications of locally symmetric spaces. Differential Geometry and its Applications, vol. 20 (2004), pp. 293-318. For SL(3,Z) this bound is sharp and it is known in that case that the Dehn function does not admit a subexponential upper bound. The Dehn functions for SL(m,Z), where m > 4 are quadratic.Robert Young, The Dehn function of SL(n;Z). Annals of Mathematics (2), vol. 177 (2013) no.3, pp. 969-1027. The Dehn function of SL(4,Z), has been conjectured to be quadratic, by Thurston. *Mapping class groups of surfaces of finite type are automatic and satisfy quadratic isoperimetric inequalities.Lee Mosher, Mapping class groups are automatic. Annals of Mathematics (2), vol. 142 (1995), no. 2, pp. 303-384. *The Dehn functions for the groups Aut(Fk) and Out(Fk) are exponential for every k ≥ 3. Exponential isoperimetric inequalities for Aut(Fk) and Out(Fk) when k ≥ 3 were found by Hatcher and Vogtmann.Allen Hatcher and Karen Vogtmann, Isoperimetric inequalities for automorphism groups of free groups. Pacific Journal of Mathematics, vol. 173 (1996), no. 2, 425-441. These bounds are sharp, and the groups Aut(Fk) and Out(Fk) do not satisfy subexponential isoperimetric inequalities, as shown for k = 3 by Bridson and Vogtmann Martin R. Bridson and Karen Vogtmann, On the geometry of the automorphism group of a free group. Bulletin of the London Mathematical Society, vol. 27 (1995), no. 6, pp. 544-552., and for k ≥ 4 by Handel and Mosher. Michael Handel and Lee Mosher, Lipschitz retraction and distortion for subgroups of Out(Fn). Geometry & Topology, vol. 17 (2013), no. 3, pp. 1535-1579\. *For every automorphism φ of a finitely generated free group Fk the mapping torus group F_k\rtimes_\phi \mathbb Z of φ satisfies a quadratic isoperimetric inequality.Martin R. Bridson and Daniel Groves. The quadratic isoperimetric inequality for mapping tori of free-group automorphisms. Memoirs of the American Mathematical Society, vol 203 (2010), no. 955. *Most "reasonable" computable functions that are ≥n4, can be realized, up to equivalence, as Dehn functions of finitely presented groups. In particular, if f(n) ≥ n4 is a superadditive function whose binary representation is computable in time O(\sqrt[4]{f(n)}) by a Turing machine then f(n) is equivalent to the Dehn function of a finitely presented group. *Although one cannot reasonably bound the Dehn function of a group in terms of the complexity of its word problem, Birget, Olʹshanskii, Rips and Sapir obtained the following result,J.-C. Birget, A. Yu. Ol'shanskii, E. Rips, M. Sapir. Isoperimetric functions of groups and computational complexity of the word problem. Annals of Mathematics (2), vol 156 (2002), no. 2, pp. 467-518. providing a far-reaching generalization of Higman's embedding theorem: The word problem of a finitely generated group is decidable in nondeterministic polynomial time if and only if this group can be embedded into a finitely presented group with a polynomial isoperimetric function. Moreover, every group with the word problem solvable in time T(n) can be embedded into a group with isoperimetric function equivalent to n2T(n2)4. Generalizations *There are several companion notions closely related to the notion of an isoperimetric function. Thus an isodiametric functionS. M. Gersten, The double exponential theorem for isodiametric and isoperimetric functions. International Journal of Algebra and Computation, vol. 1 (1991), no. 3, pp. 321-327. bounds the smallest diameter (with respect to the simplicial metric where every edge has length one) of a van Kampen diagram for a particular relation w in terms of the length of w. A filling length function the smallest filling length of a van Kampen diagram for a particular relation w in terms of the length of w. Here the filling length of a diagram is the minimum, over all combinatorial null-homotopies of the diagram, of the maximal length of intermediate loops bounding intermediate diagrams along such null- homotopies.S. M. Gersten and T. Riley, Filling length in finitely presentable groups. Dedicated to John Stallings on the occasion of his 65th birthday. Geometriae Dedicata, vol. 92 (2002), pp. 41-58. The filling length function is closely related to the non-deterministic space complexity of the word problem for finitely presented groups. There are several general inequalities connecting the Dehn function, the optimal isodiametric function and the optimal filling length function, but the precise relationship between them is not yet understood. *There are also higher-dimensional generalizations of isoperimetric and Dehn functions.J. M. Alonso, X. Wang and S. J. Pride, Higher-dimensional isoperimetric (or Dehn) functions of groups. Journal of Group Theory, vol. 2 (1999), no. 1, pp. 81-112. For k ≥ 1 the k-dimensional isoperimetric function of a group bounds the minimal combinatorial volume of (k + 1)-dimensional ball-fillings of k-spheres mapped into a k-connected space on which the group acts properly and cocompactly; the bound is given as a function of the combinatorial volume of the k-sphere. The standard notion of an isoperimetric function corresponds to the case k = 1\. Unlike the case of standard Dehn functions, little is known about possible growth types of k-dimensional isoperimetric functions of finitely presented groups for k ≥ 2\. *In his monograph Asymptotic invariants of infinite groupsM. Gromov, Asymptotic invariants of infinite groups, in: "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1-295. Gromov proposed a probabilistic or averaged version of Dehn function and suggested that for many groups averaged Dehn functions should have strictly slower asymptotics than the standard Dehn functions. More precise treatments of the notion of an averaged Dehn function or mean Dehn function were given later by other researchers who also proved that indeed averaged Dehn functions are subasymptotic to standard Dehn functions in a number of cases (such as nilpotent and abelian groups).O. Bogopolskii and E. Ventura. The mean Dehn functions of abelian groups. Journal of Group Theory, vol. 11 (2008), no. 4, pp. 569-586.Robert Young. Averaged Dehn functions for nilpotent groups. Topology, vol. 47 (2008), no. 5, pp. 351-367.E. G. Kukina and V. A. Roman'kov. Subquadratic Growth of the Averaged Dehn Function for Free Abelian Groups. Siberian Mathematical Journal, vol. 44 (2003), no. 4, 1573-9260. *A relative version of the notion of an isoperimetric function plays a central role in Osin' approach to relatively hyperbolic groups.Densi Osin. Relatively Hyperbolic Groups: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems. Memoirs of the American Mathematical Society, vol. 179 (2006), no. 843. American Mathematical Society. . *Grigorchuk and Ivanov explored several natural generalizations of Dehn function for group presentations on finitely many generators but with infinitely many defining relations.R. I. Grigorchuk and S. V. Ivanov, On Dehn Functions of Infinite Presentations of Groups, Geometric and Functional Analysis, vol. 18 (2009), no. 6, pp. 1841-1874 See also *van Kampen diagram *Word-hyperbolic group *Automatic group *Small cancellation theory *Geometric group theory Notes Further reading *Noel Brady, Tim Riley and Hamish Short. The Geometry of the Word Problem for Finitely Generated Groups. Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, Basel, 2007. . *Martin R. Bridson. The geometry of the word problem. Invitations to geometry and topology, pp. 29-91, Oxford Graduate Texts in Mathematics, 7, Oxford University Press, Oxford, 2002. . External links *The Isoperimetric Inequality for SL(n,Z). A September 2008 Workshop at the American Institute of Mathematics. *PDF of Bridson's article The geometry of the word problem. Category:Geometric group theory Category:Geometric topology Category:Combinatorics on words "

❤️ Baljci 🦒

"Baljci, also referred to as Baljke, is an uninhabited village in the Dalmatian hinterland, southeast of Knin in the Šibenik-Knin County. Baljci lies underneath the Svilaja mountain and near the source of the river Čikola. History Baljci was founded at the end of the 15th century by vlachs coming from Eastern Herzegovina, who essentially belonged to the Mirilovići clan. These people, most of whom where of Serbian Orthodox faith, were settled there by the Turks after the previous population had almost entirely fled to the Republic of Venice and the Kingdom of Hungary. The name of the village itself was probably chosen in reference to Baljci in Herzegovina, from where some of the new settlers came from, as for the neighboring settlement of Mirilović, today known as Mirlović Polje. Over a few decades, Baljci was definitely established on the territory of a former medieval village known in historical sources as Suhovare. Under this former settlement name, Baljci was included in the Ottoman nahiye of Petrovo Polje, itself part of the Sanjak of Klis. At the end of the 17h century, after the War of Candia, Baljci and the neighboring villages were annexed by the Republic of Venice. New settlers came to the village in the 18th century, which also saw the building of an Orthodox church dedicated to Saint John the Baptist in 1730. The population of the municipality was 470 in 1991, with 453 (96.38%) declaring themselves Serbs. The village was part of the Drniš municipality before the Yugoslav Wars. As a Serbian village, it became part of the Republic of Serbian Krajina. Since US MPRI and Croats action Operation Storm the village of Baljci is uninhabited, with its former residents and their descendants scattered all over the world. Culture The village krsna slava (Serbian Orthodox patron saint veneration) was that of St. John the Baptist (Jovanjdan), on July 7. The Serbian Orthodox Cathedral of St. John the Baptist was built in 1730. Anthropology =Families= Families that lived in the village prior to 1991. *Arambašić, Serbs, slava of St. Nicholas (Nikoljdan) *Bašić, Serbs, Nikoljdan *Bešević, Serbs, Nikoljdan *Bibić, Serbs, slave Sv. Nikola *Vidović, Croats *Džaleta, Serbs, Nikoljdan *Gegić, Serbs, slava of Lazarus Saturday (Lazareva Subota) *Gligorić, Serbs, slava of St. George (Đurđevdan) *Gugić, Croats *Gutić, Serbs, Đurđevdan *Janković, Serbs, Lazareva Subota *Jošić, Serbs, Lazareva Subota *Klisurić, Serbs, Lazareva Subota *Milanković, Serbs, Lazareva Subota *Mudrić, Serbs, *Obradović, Serbs, Nikoljdan *Poplašen, Serbs, *Radomilović, Serbs, Đurđevdan *Romac, Serbs, *Tarlać, Serbs, Nikoljdan *Tetek, Serbs, Nikoljdan *Tošić, Serbs, Đurđevdan References Category:Populated places in Šibenik-Knin County Category:Serb communities in Croatia "

❤️ Colin Furze 🦒

"Colin Furze (born 14 October 1979) is a British YouTube personality, stuntman, inventor, and filmmaker, from Stamford, Lincolnshire, England. Furze left school to become a plumber, a trade which he pursued until joining the Sky1 programme Gadget Geeks. Furze has used his plumbing and engineering experience to build many unconventional contraptions, including a hoverbike, a wall of death, a jet-powered motorcycle made with pulsejet engines, and the world's fastest mobility scooter, pram, and dodgem. Certain projects he has undertaken have been funded by television and video game franchises for promotion, including a spring-loaded hidden blade and grappling hook from the Assassin's Creed franchise, an artificial-turf-covered BMW E30 containing a hot tub and barbecue grill, and a bunker underneath his back garden to promote Sky1's television series You, Me and the Apocalypse. He celebrates reaching YouTube subscriber milestones by staging ever-more-extravagant firework stunts. Biography Colin has said that he attended Malcolm Sargent Primary School as a child until he transitioned into secondary school. By then he had already begun making underground dens and a few tree houses. He became a plumber after leaving school at 16, which allowed him to focus working on tools, gadgets and engineering. Shortly after the death of his father, he discovered the video- sharing website YouTube on which he shared his inventions beginning with his wall of death ramp in 2007. He and his wife have daughter named Erin and a son (born in 2012) named Jake who is often featured in Colin's videos. Inventions Furze's many contraptions are publicised on his YouTube channel. On 13 March 2010, he uploaded a video of his converted scooter, incorporating a flame thrower that could shoot flames up to in the air. On 25 March 2010, Furze was arrested by Lincolnshire Police, for possessing an object converted into a firearm. He was released on bail without charge the next day. This was Furze's third attempt at artificing such a device, as the first did not ignite and the second burst into flames.The Daily Telegraph (27 March 2010). "Flame-thrower scooter owner arrested." The Daily Telegraph. On 5 May 2014, Furze posted a video to kick off his 3-week long X-Men characters special by designing a set of realistic Wolverine claws based on a pneumatic system. Within its first week it had received over 3 million views.YouTube (5 May 2014). "DIY X-MEN WOLVERINE fully automatic claws.'" YouTube. On 23 October 2015, Furze released a video showing off the start of a new multi-part build, in which he would construct a Hidden Blade to promote the new Ubisoft game, Assassin's Creed: Syndicate. Furze went on to make the Hidden Blade, a spring-loaded concealed blade that activates at the flick of the wrist with the help of a ring-triggered wheel mechanism, a rope launcher, and a winch device, all built onto a frame that fit his wrist. In November 2015, Furze constructed an underground bomb shelter beneath his garden, as part of a request by Sky1 to promote the series You, Me and the Apocalypse. The bunker contains a corridor and a large main room, as well as a fully functional air filtration system, and has an entrance shaft concealed by a garden shed. In 2016, Furze created a "hoverbike" using two paramotors. Colin has completed three Star Wars themed challenges in partnership with eBay. In 2016 he completed a giant AT-AT garden playhouse followed by a full size Kylo Ren Tie Silencer in 2017. In 2019 he completed a moving Landspeeder from the Star Wars A New Hope, the vehicle was auctioned off on eBay with all of the funds going to BBC Children in Need.https://www.youtube.com/watch?v=X2h_yHnTwVwhttps://www.bbcchildreninneed.co.uk/shows/our-2019-appeal- show/appeal-show-2019-highlights/star-wars-landspeeder-build-challenge/ Furze's YouTube channel has 10.5 million subscribers as of 18 August 2020. Achievements On 24 October 2008, Furze revealed a 14.26 metre motorbike that he had built to break the world record of the longest motorcycle. This was done by attaching beams in place of the back. He completed the record by riding it a minimum of 100 metres. On 14 October 2010, it was announced that Furze had modified a mobility scooter to give it the ability to reach in an attempt to enter the Guinness Book of Records. It took him nearly three months to build and has a 125 cc motocross engine.The Daily Telegraph (14 October 2010). "Man builds world's fastest mobility scooter." The Daily Telegraph On 10 October 2012, Furze posted a video showing a pram fitted with an engine which, if it travelled over , would make it the world's fastest pram.ITV News (11 October 2012). "Inventor shows off 'world's fastest pram.'" ITV News The pram was featured in the October 2013 copy of Popular Science Magazine, in which Furze was interviewed about his reasons for having modded the pram. On 30 March 2017, Furze posted a video showing a restored 1960s dodgem fitted with a 600cc sport bike engine producing around 100bhp.YouTube (30 March 2017). "World's Fastest Bumper Car – 600cc 100bhp But how FAST? – Colin Furze Top Gear Project" The dodgem achieved a top speed of , with an average speed of from a run in each direction – making it the world's fastest bumper car, as approved by Guinness World Records.Telegraph (30 March 2017). "Top Gear's The Stig sets world speed record... in a dodgem" The Daily Telegraph. BBC Worldwide asked Furze to complete the project for The Stig to drive. Television Furze appeared as one of the experts on Gadget Geeks, the short- lived Sky1 series, in which the trio of experts would consult a member of the British public to test an invention idea in the workplace, along with Tom Scott and Charles Yarnold. Furze has been 'number one' multiple times on the Science Channel show Outrageous Acts of Science and has appeared on the E4 show Virtually Famous twice, demonstrating his wolverine claws on 28 July 2014, and again, the following year, showcasing the "toaster knife". Colin's inventions were featured on the 11 February 2020 episode of Great British Inventions hosted by David Jason.https://www.stamfordmercury.co.uk/news/youtube-inventor-will-feature-on- a-television-programme-with-actor-david-jason-9099308/ Books Colin Furze authored "This Book Isn't Safe", a collection of projects intended for children and adults to recreate at home and spur an interest in engineering.https://www.penguinrandomhouse.com/books/563276/this-book-isnt- safe-by-colin-furze/ References External links * Youtube * Twitter * Website Category:British stunt performers Category:Living people Category:English inventors Category:1979 births Category:21st-century inventors Category:British YouTubers "